# Para-linearity as the nonassociative counterpart of linearity

Qinghai Huo<EMAIL_ADDRESS>Department of Mathematics, University of
Science and Technology of China, Hefei 230026, China  and  Guangbin Ren
<EMAIL_ADDRESS>Department of Mathematics, University of Science and
Technology of China, Hefei 230026, China

###### Abstract.

In an octonionic Hilbert space $H$, the octonionic linearity is taken to fail
for the maps induced by the octonionic inner products, and it should be
replaced with the octonionic para-linearity. However, to introduce the notion
of the octonionic para-linearity we encounter an insurmountable obstacle. That
is, the axiom

$\left\langle pu,u\right\rangle=p\left\langle u,u\right\rangle$

for any octonion $p$ and element $u\in H$ introduced by Goldstine and Horwitz
in 1964 can not be interpreted as a property to be obeyed by the octonionic
para-linear maps. In this article, we solve this critical problem by showing
that this axiom is in fact non-independent from others. This enables us to
initiate the study of octonionic para-linear maps. We can thus establish the
octonionic Riesz representation theorem which, up to isomorphism, identifies
two octonionic Hilbert spaces with one being the dual of the other. The dual
space consists of continuous left para-linear functionals and it becomes a
right $\mathbb{O}$-module under the multiplication defined in terms of the
second associators which measures the failure of $\mathbb{O}$-linearity. This
right multiplication has an alternative expression

${(f\odot p)(x)}=pf(p^{-1}x)p,$

which is a generalized Moufang identity. Remarkably, the multiplication is
compatible with the canonical norm, i.e.,

$\left|\left|f\odot
p\right|\right|=\left|\left|f\right|\right|\left|p\right|.$

Our final conclusion is that para-linearity is the nonassociative counterpart
of linearity.

###### Key words and phrases:

Octonionic Hilbert space; para-linear; Riesz representation theorem

###### 2010 Mathematics Subject Classification:

Primary: 17A35;46S10

This work was supported by the NNSF of China (11771412).

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 $\mathbb{O}$-para-linear functions
  4. 4 $\mathbb{O}$-Hilbert spaces
  5. 5 Riesz representation theorem

##  1\. Introduction

What is the nonassociative counterpart of linearity is a natural question in
the nonassociative realm. This question has no answer even in octonionic
functional analysis since there still lacks the octonionic Riesz
representation theorem which depends on the maps abstracted from the inner
products. In order to explore the reason for this lack, we start with the
octonionic Hilbert theory.

The theory of octonionic Hilbert spaces was initiated by Goldstine and Horwitz
[3] in 1964 and has many applications in the spectral theory [13], operator
theory [12], and physics [2, 7, 6, 16]. But Compared with the quaternionic
Hilbert space theory [9, 14, 15, 21, 22], the development of the octonionic
Hilbert space theory is lagging behind. Nonetheless, the existing research on
the octonionic Hilbert space theory has emerged some effective methods. For
example, Goldstine and Horwitz [3] attribute the study of the octonionic
Hilbert space to the associative setting; see [4, 17] for the further
development. Ludkovsky and Sprössig [12, 13] find the theory can get well
developed if the $\mathbb{O}$-Hilbert space admits the tensor decomposition of
a real Hilbert space with the algebra of octonions $\mathbb{O}$.

Now we focus on the definition of the octonionic Hilbert space.

######  Definition 1.1 ([3]).

An $\mathbb{O}$-Hilbert space $H$ is a left $\mathbb{O}$-module with an
$\mathbb{O}$-inner product

$\left\langle\cdot,\cdot\right\rangle:H\times H\rightarrow\mathbb{O}$

such that $(H,\text{Re}\,\left\langle\cdot,\cdot\right\rangle)$ is a real
Hilbert space. Here the $\mathbb{O}$-inner product satisfies the following
axioms for all $u,v\in H$ and $p\in\mathbb{O}$:

  1. (a)

$\left\langle u+v,w\right\rangle=\left\langle u,w\right\rangle+\left\langle
v,w\right\rangle$;

  2. (b)

$\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}$;

  3. (c)

$\left\langle u,u\right\rangle\in\mathbb{R}^{+}$; and $\left\langle
u,u\right\rangle=0$ if and only if $u=0$;

  4. (d)

$\left\langle tu,v\right\rangle=t\left\langle u,v\right\rangle$ for
$t\in\mathbb{R}$;

  5. (e)

$\text{Re}\,\left\langle pu,v\right\rangle=\text{Re}\,(p\left\langle
u,v\right\rangle)$;

  6. (f)

$\left\langle pu,u\right\rangle=p\left\langle u,u\right\rangle$.

The functionals that need to be studied in the octonionic Hilbert space are
naturally induced by the octonionic inner product

(1.1) $\displaystyle f(x)=\langle x,v\rangle$

for any given element $v\in H$. However, these real linear maps in (1.1) are
generally not octonionic linear maps since there may hold

$f(px)\neq pf(x)$

for some $p\in\mathbb{O}$ and $x\in H$.

It is still an unsolved problem so far how to define an appropriate functional
in octonionic functional analysis. This is because in Definition 1.1 Axiom
$(f)$, as a property of $\mathbb{O}$-inner-product, cannot be abstractly
interpreted into the property of a functional on a general $\mathbb{O}$-normed
space.

Fortunately, we can overcome this obstacle in a roundabout way in this
article. We find that Axiom $(f)$ is actually non-independent. This motivates
us to introduce a kind of maps beyond $\mathbb{O}$-linearity, called
$\mathbb{O}$-para-linear maps, defined in any $\mathbb{O}$-modules.

An $\mathbb{O}$-para-linear map in a left $\mathbb{O}$-module $M$ is defined
to be a real linear map $f\in\text{Hom}_{\mathbb{R}}(M,\mathbb{O})$ subject to
the condition

(1.2) $\displaystyle\text{Re}\,[p,x,f]=0$

for all $p\in\mathbb{O}$ and $x\in M$, where the element

$[p,x,f]:=f(px)-pf(x)$

is called the second associator related to $f$. Condition (1.2) becomes Axiom
(e) in Definition 1.1 in the specific case where $f$ is a map induced by the
octonionic inner product as in (1.1).

The set of para-linear functions in a left $\mathbb{O}$-module constitutes a
right $\mathbb{O}$-module under the multiplication

(1.3) $\displaystyle(f\odot p)(x):=f(x)p-[p,x,f].$

Similarly, $\mathbb{K}$-para-linearity can also be defined for any real normed
division algebra

$\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O},$

where $\mathbb{H}$ is the algebra of quaternions. Since $\mathbb{K}$-para-
linearity degenerates to linearity in the associative setting, this shows
para-linearity is a nonassociative counterpart of linearity.

$\mathbb{K}$-para-linearity can be used to define $\mathbb{K}$-Hilbert spaces.
In the specific case where $\mathbb{K}=\mathbb{O}$, this definition is
belonging to Goldstine and Horwitz [3] except that a non-independent axiom is
removed.

######  Definition 1.2.

Suppose that $H$ is a real Hilbert space under the real inner product
$\langle\cdot,\cdot\rangle_{\mathbb{R}}$ and also a left $\mathbb{K}$-module.
We call $H$ a  $\mathbb{K}$-Hilbert space if there exists an
$\mathbb{R}$-bilinear map

$\left\langle\cdot,\cdot\right\rangle:H\times H\rightarrow\mathbb{K}$

such that

$Re\left\langle\cdot,\cdot\right\rangle=\langle\cdot,\cdot\rangle_{\mathbb{R}}$

and satisfying

  1. (${\mathrm{i}}$)

($\mathbb{K}$-para-linearity) $\left\langle\cdot,u\right\rangle$ is left
$\mathbb{K}$-para-linear for all $u\in H$.

  2. (${\mathrm{ii}}$)

($\mathbb{K}$-hermiticity) $\left\langle
u,v\right\rangle=\overline{\left\langle v,u\right\rangle}$ for all $u,v\in H$.

  3. (${\mathrm{iii}}$)

(Positivity) $\left\langle u,u\right\rangle\in\mathbb{R}^{+}$; and
$\left\langle u,u\right\rangle=0$ if and only if $u=0$.

Now we can use $\mathbb{O}$-para-linearity to study the Riesz representation
theorem in $\mathbb{O}$-Hilbert spaces $H$. The theorem should be expressed in
the following form

(1.4) $\displaystyle H^{*}\cong{H}.$

Here $H^{*}$ is the dual space consisting of $\mathbb{O}$-para-linear
functions and the bijection in (1.4) between two spaces is both an isometric
isomorphism as real Banach spaces and a conjugate isomorphism as
$\mathbb{O}$-modules.

Of the most difficult is to show the compatibility between the module
structure of $H^{*}$ and its canonical norm, i.e.,

$\left|\left|f\odot p\right|\right|=\left|\left|f\right|\right|\left|p\right|$

for all $f\in H^{*}$ and $p\in\mathbb{O}$, where $f\odot p$ is the right
multiplication defined in (1.3). Overcoming this difficulty requires another
expression of the right multiplication

${(f\odot p)(x)}=pf(p^{-1}x)p,$

which is a generalized Moufang identity [18].

The octonionic Riesz representation theorem in (1.4) can be restated as

(1.5) $\displaystyle(H^{*})^{-}\cong{H}.$

Here $(H^{*})^{-}$ denotes the set $H^{*}$ endowed with a left
$\mathbb{O}$-module defined by

$p\cdot x:=x\overline{p}.$

The point is that the bijection in (1.5) is an isomorphism of two left
$\mathbb{O}$-modules in contrast to the conjugate isomorphism in (1.4).

Space $(H^{*})^{-}$ can be endowed with an $\mathbb{O}$-inner product via
isomorphism (1.5) so that it becomes an $\mathbb{O}$-Hilbert space. The
octonionic Riesz representation theorem shows that the two
$\mathbb{O}$-Hilbert spaces in (1.5) coincide up to isomorphism. This result
is true if $\mathbb{O}$ is replaced by any normed division algebras
$\mathbb{K}$.

######  Theorem 1.3 (Riesz representation theorem).

Let $\mathbb{K}$ be a normed division algebra. Let $H$ be a
$\mathbb{K}$-Hilbert space and $H^{*}$ its dual consisting of all continuous
$\mathbb{K}$-para-linear functionals. Then we can identify the two
$\mathbb{K}$-Hilbert spaces $H$ and $(H^{*})^{-}$ up to isomorphism.

Opposed to the associative setting, we have two distinct spaces

$H^{*_{\mathbb{O}}}\subsetneq H^{*}.$

Here $H^{*_{\mathbb{O}}}$ consists of all continuous $\mathbb{O}$-linear
functionals of $H$. Under isomorphism (1.4), $H^{*_{\mathbb{O}}}$ is
corresponding to the nucleus of $H$ consisting of associative elements.
Symbolically,

$\displaystyle H^{*_{\mathbb{O}}}\cong\mathscr{A}(H).$

This shows that the class of $\mathbb{O}$-linear functionals is too small to
represent every element in an $\mathbb{O}$-Hilbert space.

We remark that there are profound relations between the associative and
nonassociative theories. For example, the algebra of octonions is closely
related to the Clifford algebra of $6,7$ dimensions [8], and the category of
left $\mathbb{O}$-modules is shown to be isomorphic to the category of left
$C\ell_{7}$-modules [10]. But the octonionic theory has its intrinsic objects
such as the $\mathbb{O}$-para-linear functionals as demonstrated in the
octonionic Riesz representation theorem.

After para-linearity is introduced as a substitute for linearity in
nonassociative framework, one can further study para-linear operators, weak
topology related to para-linear functionals, octonionic $C^{*}$ algebra, and
para-linear connections.

##  2\. Preliminaries

In this section we review some notations and basic properties about octonions
$\mathbb{O}$ and left $\mathbb{O}$-modules. We refer to [1, 10] for more
details.

###  2.1. Octonions

The set of octonions $\mathbb{O}$ is a nonassociative, noncommutative, normed
division algebra over $\mathbb{R}$. Let

$e_{0}=1,\ e_{1},\ \dots,\ e_{7}$

be a basis of $\mathbb{O}$ as a real vector space subject to the
multiplication rule

(2.1) $\displaystyle
e_{i}e_{j}=\sum_{k=1}^{7}\epsilon_{ijk}e_{k}-\delta_{ij}.$

for any $i,j=1,\dots,7$. Here $\delta_{ij}$ is the Kronecker delta and
$\epsilon_{ijk}$ is completely antisymmetric with value 1 precisely when

$ijk=123,145,176,246,257,347,365.$

Let $x=x_{0}+\sum_{i=1}^{7}x_{i}e_{i}$ be an octonion with all
$x_{i}\in\mathbb{R}$. We define its conjugate

$\overline{x}:=x_{0}-\sum_{i=1}^{7}x_{i}e_{i},$

its norm

$|x|=\sqrt{x\overline{x}},$

and its real part

$\text{Re}\,{x}=x_{0}=\frac{1}{2}(x+\overline{x}).$

###  2.2. $\mathbb{O}$-modules

We recall some basic notations and results on $\mathbb{O}$-modules in this
subsection. We refer to [11, 20, 19, 10] for octonionic modules and even for
modules of alternative algebras.

A left $\mathbb{O}$-module $M$ is a real vector space $M$ endowed with an
$\mathbb{O}$-scalar multiplication such that the left associator is left
alternative, i.e., for all $p,q\in\mathbb{O}$ and $m\in M$ we have

(2.2) $\displaystyle[p,q,m]=-[q,p,m].$

Here the left associator is defined by

$[p,q,m]:=(pq)m-p(qm).$

Condition (2.2) is equivalent to the requirement

$r(rm)=r^{2}m$

for all $r\in\mathbb{O}$ and all $m\in M$.

Of the most important identity involving associators in a left
$\mathbb{O}$-module $M$ is the five-term identity

(2.3) $[p,q,r]m+p[q,r,m]=[pq,r,m]-[p,qr,m]+[p,q,rm]$

for all $p,q,r\in\mathbb{O}$ and $m\in M$. This even holds when $M$ is a
nonassociative algebra [18].

In the theory of left $\mathbb{O}$-modules, associative and conjugate
associative elements play key roles ([10]). Let $M$ be a left
$\mathbb{O}$-module. An element $m\in M$ is called an associative element if

$(pq)m=p(qm)$

and called a conjugate associative element if

$(pq)m=q(pm).$

The set of all associative elements is called the nucleus of $M$ and denoted
by $\mathscr{A}(M)$, i.e.,

$\mathscr{A}(M):=\\{m\in M\mid[p,q,m]=0\text{ for all }p,q\in\mathbb{O}\\}.$

The set of all conjugate associative elements of $M$ is denoted by
$\mathscr{A}^{-}(M)$, i.e.,

$\mathscr{A}^{-}(M):=\\{m\in M\mid(pq)m=q(pm)\text{ for all
}p,q\in\mathbb{O}\\}.$

######  Theorem 2.1 ([10]).

Let $M$ be a left $\mathbb{O}$-module. Then

$M\cong\mathbb{O}\mathscr{A}(M)\oplus{\mathbb{O}}\mathscr{A}^{-}(M).$

##  3\. $\mathbb{O}$-para-linear functions

The notion of the second associator is used to describe the extent to which a
real linear map is far away from the octonionic linear map. This notion also
allows us to define the para-linear map.

######  Definition 3.1.

Let $M$ be a left $\mathbb{O}$-module and let
$\text{Hom}_{\mathbb{R}}(M,\mathbb{O})$ be the set of all real linear
functions $f:M\to\mathbb{O}$. We introduce a real trilinear map

$[\cdot,\cdot,\cdot]:\mathbb{O}\times
M\times\text{Hom}_{\mathbb{R}}(M,\mathbb{O})\xlongrightarrow[]{}\mathbb{O},$

defined by

(3.1) $[p,x,f]:=f(px)-pf(x).$

We call this map the second associator in $M$, which measures the failure of
$\mathbb{O}$-linearity.

######  Definition 3.2.

Let $M$ be a left $\mathbb{O}$-module. A real linear map
$f:M\xlongrightarrow[]{}\mathbb{O}$ is called an $\mathbb{O}$-para-linear
function if it satisfies

$\text{Re}\,[p,x,f]=0$

for any $p\in\mathbb{O}$ and $x\in M$.

We denote by $\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})$ the set of all
$\mathbb{O}$-para-linear functions. By definition, we have

$\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})=\\{f\in\text{Hom}_{\mathbb{R}}(M,\mathbb{O}):\
\text{Re}\,[p,x,f]=0\ \ \mbox{for \ all \ $p\in\mathbb{O}$ \ and \ $x\in M$
}\\}.$

######  Example 3.3.

The right multiplication on $\mathbb{O}$ provides a typical example of
$\mathbb{O}$-para-linear function, i.e.,

$R_{q}\in\text{Hom}_{L\mathbb{O}}(\mathbb{O},\mathbb{O})$

for any $q\in\mathbb{O}$. Here $R_{q}$ denotes the right multiplication
defined by

$R_{q}(x)=xq.$

In this typical example, we find that the associator and the second associator
coincide, i.e.,

(3.2) $[p,x,R_{q}]=[p,x,q]$

for any $p,x,q\in\mathbb{O}$. Indeed, by definition we have

$[p,x,R_{q}]=R_{q}(px)-pR_{q}(x)=(px)q-p(xq)=[p,x,q].$

Identity (3.2) also implies

$\text{Re}\,[p,x,R_{q}]=0$

so that $R_{q}$ is $\mathbb{O}$-para-linear.

If $f\in\text{Hom}_{\mathbb{R}}(M,\mathbb{O})$, then it can be written as

(3.3) $\displaystyle f(x)=f_{\mathbb{R}}(x)+\sum_{i=1}^{7}e_{i}f_{i}(x),$

where

$f_{\mathbb{R}}(x):=\text{Re}\,f(x)$

and $f_{i}(x)\in\mathbb{R}\text{ for }i=1,\dots,7$.

Now we give a criterion for determining whether a real linear function
$f\in\text{Hom}_{\mathbb{R}}(M,\mathbb{O})$ is an $\mathbb{O}$-para-linear
map.

######  Theorem 3.4.

Let $f\in\text{Hom}_{\mathbb{R}}(M,\mathbb{O})$ with expression as in (3.3).
Then $f$ is $\mathbb{O}$-para-linear if and only if

(3.4) $\displaystyle f_{i}(x)=f_{\mathbb{R}}(\overline{e_{i}}x)$

for all $i=0,\dots,7$ and all $x\in\mathbb{O}$.

###### Proof.

We assume $f$ $\mathbb{O}$-para-linear. By definition, we have

$\displaystyle f_{\mathbb{R}}(\overline{e_{i}}x)$
$\displaystyle=\text{Re}\,f(\overline{e_{i}}x)$
$\displaystyle=\text{Re}\,(\overline{e_{i}}f(x))$
$\displaystyle=\text{Re}\,\Big{(}\overline{e_{i}}\sum_{j=0}^{7}e_{j}f_{j}(x)\Big{)}$
$\displaystyle=f_{i}(x).$

Conversely, under assumptions (3.4), for any $i=1,\ldots,7$ we claim

(3.5) $\displaystyle
e_{i}f(x)=e_{i}f_{\mathbb{R}}(x)+\sum_{j,k=1}^{7}\epsilon_{ijk}e_{k}f_{j}(x)-f_{i}(x)$

Indeed, by calculation we have

$\displaystyle e_{i}f(x)$ $\displaystyle=$ $\displaystyle
e_{i}(f_{\mathbb{R}}(x)+\sum_{j=1}^{7}e_{j}f_{j}(x))$ $\displaystyle=$
$\displaystyle e_{i}f_{\mathbb{R}}(x)+\sum_{j=1}^{7}e_{i}e_{j}f_{j}(x)$
$\displaystyle=$ $\displaystyle
e_{i}f_{\mathbb{R}}(x)+\sum_{j,k=1}^{7}\epsilon_{ijk}e_{k}f_{j}(x)-f_{i}(x)$

and the claim holds. From the claim we have

$\displaystyle\text{Re}\,[e_{i},x,f]$ $\displaystyle=$ $\displaystyle
Re(f(e_{i}x)-e_{i}f(x))$ $\displaystyle=$ $\displaystyle
f_{\mathbb{R}}(e_{i}x)+f_{i}(x)=0.$

This finishes the proof.

∎

The second associator of a $\mathbb{O}$-para-linear map in
$\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})$ has a close relation with the
associator in $\mathbb{O}$.

######  Theorem 3.5.

Let $f\in\text{Hom}_{\mathbb{R}}(M,\mathbb{O})$ with expression as in (3.3).
Then $f$ is $\mathbb{O}$-para-linear if and only if

(3.6) $\displaystyle[p,x,f]=\sum_{i=1}^{7}e_{i}f_{\mathbb{R}}([e_{i},p,x])$

for all $i=0,\dots,7$ and all $p\in\mathbb{O}$ and $x\in M$.

###### Proof.

To prove the necessity, we might as well set $p=e_{i}$ with $i=1,\ldots,7$.
Since $f$ is $\mathbb{O}$-para-linear, Theorem 3.4 shows

(3.7) $\displaystyle
f(x)=f_{\mathbb{R}}(x)-\sum_{j=1}^{7}e_{j}f_{\mathbb{R}}(e_{j}x).$

Notice that

(3.8) $\displaystyle
e_{j}(e_{i}x)=(e_{j}e_{i})x-[e_{j},e_{i},x]=\sum_{k=1}^{7}\epsilon_{jik}e_{k}x-\delta_{ij}x-[e_{j},e_{i},x].$

Now we replace $x$ by $e_{i}x$ in (3.7) and then apply (3.8). Due to the real
linearity of $f_{\mathbb{R}}$ we have

(3.9) $\displaystyle
f(e_{i}x)=f_{\mathbb{R}}(e_{i}x)-\sum_{j,k=1}^{7}\epsilon_{jik}e_{j}f_{\mathbb{R}}(e_{k}x)+e_{i}f_{\mathbb{R}}(x)+\sum_{j=1}^{7}e_{j}f_{\mathbb{R}}([e_{j},e_{i},x]).$

On the other hand we have shown in (3.5) that

(3.10) $\displaystyle
e_{i}f(x)=e_{i}f_{\mathbb{R}}(x)+\sum_{j,k=1}^{7}\epsilon_{ijk}e_{k}f_{j}(x)-f_{i}(x)$

We subtract both sides in (3.9) and (3.10). Since $\epsilon_{ijk}$ is
completely antisymmetric, we apply the fact that

$f_{i}(x)=-f_{\mathbb{R}}({e_{i}}x)$

and identity (3.4) to conclude

$\displaystyle[e_{i},x,f]=f(e_{i}x)-e_{i}f(x)=\sum_{j=1}^{7}e_{j}f_{\mathbb{R}}([e_{j},p,x]).$

Conversely, if conditions (3.6) hold, then

$\text{Re}\,[p,x,f]=0$

so that $f$ is $\mathbb{O}$-para-linear by definition. ∎

######  Remark 3.6.

Theorems 3.4 and 3.5 hold if $\mathbb{O}$ is replaced by any Cayley-Dickson
algebra $\mathbb{K}$ (see [8]) under the similar notion of $\mathbb{K}$-para-
linear functions. $\mathbb{K}$-para-linearity becomes $\mathbb{K}$-linearity
when $\mathbb{K}$ is an associative normed division algebra because the
associators in (3.6) vanish in this setting. As a result, the notion of para-
linearity is a canonical generalization of the classical linearity from the
associative to the nonassociative realm.

As direct consequences of Theorems 3.4 and 3.5, the second associator vanishes
for associative element and an $\mathbb{O}$-para-linear map is uniquely
determined by its real part.

######  Corollary 3.7.

Assume that $f:M\to\mathbb{O}$ is $\mathbb{O}$-para-linear.

  1. (${\mathrm{i}}$)

For each associative element $x\in\mathscr{A}(M)$, we have

$[p,x,f]=0$

and hence for all $p\in\mathbb{O}$

$f(px)=pf(x).$

  2. (${\mathrm{ii}}$)

$f_{\mathbb{R}}=0$ if and only if $f=0$.

To study the $\mathbb{O}$-module structure of
$\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})$, we need some useful identities about
the second associators.

######  Proposition 3.8.

Suppose that $f\in\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})$. Then for any
$p,q\in\mathbb{O}$ and $x\in M$ we have

(3.11) $\displaystyle p[p,x,f]$
$\displaystyle=[p,x,f]\overline{p}=[p,\overline{p}x,f];$ (3.12)
$\displaystyle\text{Re}\,\left(f([p,q,x])\right)$
$\displaystyle=\text{Re}\,\left([p,x,f]{q}\right);$ (3.13)
$\displaystyle\text{Re}\,\left([p,x,f]q\right)$
$\displaystyle=-\text{Re}\,\left([q,x,f]p\right).$

###### Proof.

As usual, we write

$f(x)=f_{\mathbb{R}}(x)+\sum_{i=1}^{7}e_{i}f_{i}(x).$

To prove (3.11), without loss of generality we may assume that
$p\in\mathit{Im}(\mathbb{O})$. Let $p=\sum_{i=1}^{7}p_{i}e_{i}$ with
$p_{i}\in\mathbb{R}$. Then by Theorem 3.5, we have

$\displaystyle p[p,x,f]$
$\displaystyle=\sum_{i,j=1}^{7}p_{i}p_{j}e_{i}[e_{j},x,f]$
$\displaystyle=\sum_{i,j,k=1}^{7}p_{i}p_{j}e_{i}e_{k}f_{\mathbb{R}}([e_{k},e_{j},x]).$

Similarly, we get

$\displaystyle[p,x,f]\overline{p}$
$\displaystyle=\sum_{i,j=1}^{7}p_{i}p_{j}[e_{j},x,f]\overline{e_{i}}$
$\displaystyle=-\sum_{i,j,k=1}^{7}p_{i}p_{j}e_{k}e_{i}f_{\mathbb{R}}([e_{k},e_{j},x]).$

Since

$e_{i}e_{k}+e_{k}e_{i}=-2\delta_{ik},$

it follows that

$p[p,x,f]-[p,x,f]\overline{p}=-2\sum_{i,j=1}^{7}p_{i}p_{j}f_{\mathbb{R}}([e_{i},e_{j},x]).$

Because of the skew symmetricity of the associator, we get

$p[p,x,f]=[p,x,f]\overline{p}.$

By direct calculations, we get

$\displaystyle[{p},\overline{p}x,f]$ $\displaystyle=$ $\displaystyle
f(\left|p\right|^{2}x)-pf({\overline{p}}x)$ $\displaystyle=$
$\displaystyle\left|p\right|^{2}f(x)-pf({\overline{p}}x)$ $\displaystyle=$
$\displaystyle p(\overline{p}f(x))-pf({\overline{p}}x)$ $\displaystyle=$
$\displaystyle-p[\overline{p},x,f]$ $\displaystyle=$ $\displaystyle
p[{p},x,f].$

Now we come to prove (3.12). without loss of generality we may assume
$q=e_{i}$ with $i=1,\dots,7$. By direct calculations, we have

$\displaystyle\text{Re}\,\left([p,x,f]{q}\right)$
$\displaystyle=\text{Re}\,\Big{(}\sum_{j=1}^{7}(e_{j}f_{\mathbb{R}}([e_{j},p,x]))e_{i}\Big{)}$
$\displaystyle=\text{Re}\,\Big{(}\sum_{j=1}^{7}f_{\mathbb{R}}([e_{j},p,x])e_{j}e_{i}\Big{)}$
$\displaystyle=-f_{\mathbb{R}}([e_{i},p,x])$
$\displaystyle=\text{Re}\,\big{(}f([p,q,x])\big{)}.$

Identity (3.13) follows from (3.12) directly. ∎

######  Remark 3.9.

Identity (3.11) generalizes the third identity of Lemma 3.9 in [5].

Utilizing Proposition 3.8, we can endow
$\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})$ with a canonical right
$\mathbb{O}$-module structure. Let $r\in\mathbb{O}$,
$f\in\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})$, and $x\in M$. We define the
right scalar multiplication

(3.14) $\displaystyle(f\odot r)(x)\coloneqq\ f(x)r-[r,x,f].$

######  Theorem 3.10.

If $M$ be a left $\mathbb{O}$-module, then
$\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})$ is a right $\mathbb{O}$-module.

###### Proof.

First we need to show that for any
$f\in\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})$ and $r\in\mathbb{O}$ we have

$f\odot r\in\text{Hom}_{L\mathbb{O}}(M,\mathbb{O}).$

That is to prove

(3.15) $\displaystyle\text{Re}\,[p,x,f\odot r]=0.$

By direct calculations, we have

$\displaystyle[p,x,f\odot r]$ $\displaystyle=(f\odot r)(px)-p((f\odot r)(x))$
$\displaystyle=f(px)r-[r,px,f]-p\left(f(x)r-[r,x,f]\right)$
$\displaystyle=\left(pf(x)+[p,x,f]\right)r-[r,px,f]-p\left(f(x)r-[r,x,f]\right)$
(3.16) $\displaystyle=[p,f(x),r]+[p,x,f]r-[r,px,f]+p[r,x,f].$

Since

$\text{Re}\,[p,f(x),r]=0,\quad\text{Re}\,[r,px,f]=0,$

(3) tells us that we have to show

$\text{Re}\,([p,x,f]r)=-\text{Re}\,(p[r,x,f]).$

This can be easily deduced by applying identity (3.13) and the fact that

$\text{Re}\,(pq)=\text{Re}\,(qp)$

for all $p,q\in\mathbb{O}$.

Next we come to show that $\text{Hom}_{L\mathbb{O}}(M,\mathbb{O})$ is a right
$\mathbb{O}$-module. It suffices to prove

$(f\odot r)\odot r=f\odot(r^{2})$

for any $r\in\mathbb{O}$. Due to the uniqueness as shown in assertion
$({\mathrm{ii}})$ of Corollary 3.7, we need to prove

$\text{Re}\,{\big{(}(f\odot r)\odot r-f\odot(r^{2})\big{)}(x)}=0.$

By definition (3.14) and identity (3.11), we have

$\displaystyle{((f\odot r)\odot r)(x)}$ $\displaystyle={\big{(}(f\odot
r)(x)\big{)}}r-[r,x,f\odot r]$ (3.17)
$\displaystyle=f(x)r^{2}-[r,x,f]r-[r,x,f\odot r]$ and (3.18)
$\displaystyle{(f\odot(r^{2}))(x)}$ $\displaystyle=f(x)r^{2}-[{r^{2}},x,f].$

Notice that $\text{Re}\,[r,x,f\odot r]=0$, $\text{Re}\,[{r^{2}},x,f]=0$, and

(3.19) $\displaystyle[r,x,f]=-[\overline{r},x,f].$

We thus conclude from (3.17) and (3.18) that

$\displaystyle\text{Re}\,{\big{(}(f\odot r)\odot r-f\odot(r^{2})\big{)}(x)}$
$\displaystyle=\text{Re}\,([\overline{r},x,f]r)=\text{Re}\,([\overline{r},rx,f])=0.$

The last but one step used (3.11). This completes the proof. ∎

##  4\. $\mathbb{O}$-Hilbert spaces

We adopt the definition of $\mathbb{O}$-Hilbert spaces by Goldstine and
Horwitz [3] except that we describe it in terms of $\mathbb{O}$-para-
linearity. In addition, we find a non-independent axiom given by Goldstine and
Horwitz [3].

###  4.1. Definition of Hilbert $\mathbb{O}$-modules

We first introduce a succinct definition of $\mathbb{O}$-Hilbert spaces. This
definition also suits all normed division algebras.

######  Definition 4.1.

A left $\mathbb{O}$-module $H$ is called a  pre-Hilbert left
$\mathbb{O}$-module if there exists an $\mathbb{R}$-bilinear map
$\left\langle\cdot,\cdot\right\rangle:H\times H\rightarrow\mathbb{O}$,
referred to as an $\mathbb{O}$-inner product, satisfying:

  1. (${\mathrm{i}}$)

($\mathbb{O}$-para-linearity) $\left\langle\cdot,u\right\rangle$ is left
$\mathbb{O}$-para-linear for all $u\in H$.

  2. (${\mathrm{ii}}$)

(Octonion hermiticity) $\left\langle u,v\right\rangle=\overline{\left\langle
v,u\right\rangle}$ for all $u,v\in H$.

  3. (${\mathrm{iii}}$)

(Positivity) $\left\langle u,u\right\rangle\in\mathbb{R}^{+}$; and
$\left\langle u,u\right\rangle=0$ if and only if $u=0$.

Under axioms in Definition 4.1, we can deduce in Lemma 4.6 below that

(4.1) $\left\langle pu,u\right\rangle=p\left\langle u,u\right\rangle$

for all $p\in\mathbb{O}$ and all $u\in H$. This equation is one of the axioms
of Goldstine and Horwitz as part $(f)$ of [3, Postulate $2$].

We introduce a real trilinear map

$[\cdot,\cdot,\cdot]:\mathbb{O}\times M\times
M\xlongrightarrow[]{}\mathbb{O},$

defined by

$[p,u,v]:=\left\langle pu,v\right\rangle-p\left\langle u,v\right\rangle$

for $u,v\in H$ and $p\in\mathbb{O}$. We call this map the second associator in
$H$.

Identity (4.1) can be restated as

$[p,u,u]=0.$

######  Remark 4.2.

The second associator has been introduced in the space
$\text{Hom}_{L\mathbb{O}}(H,\mathbb{O})$ in Section 3. These two notions in
$H$ and $H^{*}$ coincide if we identify $H$ with $H^{*}$ (see Theorem 5.6). In
fact, for any $p\in\mathbb{O}$ and $u,v\in H$ we have

(4.2) $\displaystyle[p,u,v]=[p,u,v^{\prime}],$

where $v^{\prime}\in{\text{Hom}_{L\mathbb{O}}(H,\mathbb{O})}$ is a map
associated to $v$ via the inner product, i.e.,

$v^{\prime}=\left\langle\cdot,v\right\rangle.$

Indeed, by definition we have

$\displaystyle[p,u,v^{\prime}]$ $\displaystyle=$ $\displaystyle
v^{\prime}(pu)-pv^{\prime}(u)$ $\displaystyle=$ $\displaystyle\left\langle
pu,v\right\rangle-p\left\langle u,v\right\rangle$ $\displaystyle=$
$\displaystyle[p,u,v].$

######  Example 4.3.

If $\mathbb{O}$ is regarded as an $\mathbb{O}$-Hilbert space with the inner
product

$\left<x,y\right>:=x\overline{y}$

then its second associator of $p,u,v$ is given by

$\displaystyle[p,u,v]$ $\displaystyle=$
$\displaystyle\left<pu,v\right>-p\left<u,v\right>$ $\displaystyle=$
$\displaystyle(pu)\overline{v}-p(u\overline{v}),$

which is exactly the associator of $p,u,\overline{v}$ in the algebra
$\mathbb{O}$.

Second associators in an $\mathbb{O}$-Hilbert space can be studied with the
aid of the theory of $\mathbb{O}$-para-linear functionals based on relation
(4.2).

######  Lemma 4.4.

Let $H$ be a pre-Hilbert (left) $\mathbb{O}$-module, $p,q\in\mathbb{O}$ and
$v,u\in H$. Then we have

(4.3) $\displaystyle[p,v,u]$
$\displaystyle=\sum_{i=1}^{7}e_{i}\text{Re}\,\left<[e_{i},p,v],u\right>;$
(4.4) $\displaystyle[{pq},v,u]$
$\displaystyle=\left<[p,q,v],u\right>-[p,q,\left<v,u\right>]+p[q,v,u]+[p,qv,u];$
(4.5) $\displaystyle\text{Re}\,\left<[p,q,v],u\right>$
$\displaystyle=-\text{Re}\,\left(p[q,v,u]\right).$

###### Proof.

We fix $u\in H$ and take
$f:=u^{\prime}\in{\text{Hom}_{L\mathbb{O}}(H,\mathbb{O})}$ in Theorem 3.5.
Since

$f(x)=u^{\prime}(x)=\langle x,u\rangle$

and

$f_{\mathbb{R}}(x)=\text{Re}\,\langle x,u\rangle,$

we thus conclude that identity (4.3) follows from Theorem 3.5.

Identity (4.4) can be proved in the manner similar to the proof of identity
(2.3). Identity (4.5) follows from (4.4) by taking the real part on both
sides. ∎

If $H$ is a pre-Hilbert (left) $\mathbb{O}$-module, then
$(H,\left\langle\cdot,\cdot\right\rangle_{\mathbb{R}})$ is a real Hilbert
space, where

$\left\langle x,y\right\rangle_{\mathbb{R}}:=\text{Re}\,\left\langle
x,y\right\rangle.$

Conversely, it is quite natural to ask if a real Hilbert space with a
structure of left $\mathbb{O}$-module admits an $\mathbb{O}$-inner product.

######  Theorem 4.5.

Let $H$ be a left $\mathbb{O}$-module equipped with a real valued inner
product $\left<\cdot,\cdot\right>_{0}$. Then $H$ admits an $\mathbb{O}$-inner
product $\left<\cdot,\cdot\right>$ such that

(4.6) $\left<\cdot,\cdot\right>_{\mathbb{R}}=\left<\cdot,\cdot\right>_{0}$

if and only if

(4.7) $\left\langle px,y\right\rangle_{0}=\left\langle
x,\overline{p}y\right\rangle_{0}$

holds for all $p\in\mathbb{O}$ and all $x,y\in H$.

###### Proof.

Suppose the $\mathbb{O}$-inner product $\left<\cdot,\cdot\right>$ in $H$
satisfies (4.6). To prove (4.7), we need to show

$\text{Re}\,\left<px,y\right>=\text{Re}\,\left\langle
x,\overline{p}y\right\rangle$

for all $p\in\mathbb{O}$ and $x,y\in H$. By direct calculation, we have

$\displaystyle\text{Re}\,\left<px,y\right>$ $\displaystyle=$
$\displaystyle\text{Re}\,(p\left<x,y\right>+[p,x,y])$ $\displaystyle=$
$\displaystyle\text{Re}\,(p\left<x,y\right>)$ $\displaystyle=$
$\displaystyle\text{Re}\,(\overline{p\left<x,y\right>})$ $\displaystyle=$
$\displaystyle\text{Re}\,({\left<y,x\right>}\overline{p})$ $\displaystyle=$
$\displaystyle\text{Re}\,(\overline{p}{\left<y,x\right>})$ $\displaystyle=$
$\displaystyle\text{Re}\,{\left<\overline{p}y,x\right>}$ $\displaystyle=$
$\displaystyle\text{Re}\,\left\langle x,\overline{p}y\right\rangle.$

Conversely, suppose that (4.7) holds. We claim that the following inner
product satisfies all conditions:

(4.8) $\displaystyle\left\langle x,y\right\rangle_{\mathbb{O}}:=\left\langle
x,y\right\rangle_{0}-\sum_{i=1}^{7}\left\langle
e_{i}x,y\right\rangle_{0}e_{i}.$

According to the construction, equation (4.6) obviously holds. It remains to
show that $\left\langle\cdot,\cdot\right\rangle_{\mathbb{O}}$ is an
$\mathbb{O}$-inner product.

Evidently, $\left\langle\cdot,\cdot\right\rangle_{\mathbb{O}}$ is
$\mathbb{R}$-bilinear. We come to prove the para-linearity. Define

(4.9) $f(x):=\left<x,y\right>_{\mathbb{O}}$

for any given $y\in H$. Then $f$ is clearly a real linear map. We write

(4.10) $f(x)=f_{\mathbb{R}}(x)+\sum_{i=1}^{7}f_{i}(x)e_{i},$

where $f_{\mathbb{R}}(x),\,f_{i}(x)\in\mathbb{R}$. Combining (4.8) with (4.9),
we obtain another expression of $f$ which demonstrates that

$f_{i}(x)=-\left\langle e_{i}x,y\right\rangle_{0}=-\text{Re}\,\left\langle
e_{i}x,y\right\rangle=-\text{Re}\,f(e_{i}x)=-f_{\mathbb{R}}(e_{i}x)$

for $i=1,\dots,7$. Then Theorem 3.4 implies that $f$ is left para-linear. This
proves the para-linearity of
$\left\langle\cdot,\cdot\right\rangle_{\mathbb{O}}$.

The octonionic hermiticity follows from condition (4.7). Indeed,

$\displaystyle\overline{{\left<x,y\right>}_{\mathbb{O}}}$
$\displaystyle=\left\langle x,y\right\rangle_{0}+\sum_{i=1}^{7}\left\langle
e_{i}x,y\right\rangle_{0}e_{i}$ $\displaystyle=\left\langle
y,x\right\rangle_{0}+\sum_{i=1}^{7}\left\langle
x,\overline{e_{i}}y\right\rangle_{0}e_{i}$ $\displaystyle=\left\langle
y,x\right\rangle_{0}-\sum_{i=1}^{7}\left\langle{e_{i}}y,x\right\rangle_{0}e_{i}$
$\displaystyle=\left<y,x\right>_{\mathbb{O}}.$

Finally, we show the positivity of
$\left\langle\cdot,\cdot\right\rangle_{\mathbb{O}}$. For each $i=1,\dots,7$,
we have

$\left<e_{i}x,x\right>_{0}=\left<x,\overline{e_{i}}x\right>_{0}=-\left<x,e_{i}x\right>_{0}=-\left<e_{i}x,x\right>_{0}$

so that $\left<e_{i}x,x\right>_{0}=0$. This means

$\left<x,x\right>_{\mathbb{O}}=\left<x,x\right>_{0},$

which implies the positivity. ∎

The associator $[p,q,u]$ in the $\mathbb{O}$-Hilbert space $H$ is an skew-
adjoint map as a function of $u$, while the second associator $[p,u,v]$ in the
$\mathbb{O}$-Hilbert space $H$ is an alternating real bilinear map of $u$ and
$v$.

######  Lemma 4.6.

Let $H$ be a pre-Hilbert left $\mathbb{O}$-module. For any $u,v\in H$ and
$p,q\in\mathbb{O}$, we have

(4.11) $\displaystyle\left\langle[p,q,u],v\right\rangle_{\mathbb{R}}$
$\displaystyle=$ $\displaystyle-\left\langle
u,[p,q,v]\right\rangle_{\mathbb{R}};$ (4.12) $\displaystyle{[p,u,v]}$
$\displaystyle=$ $\displaystyle-[p,v,u].$

###### Proof.

Identity (4.11) follows from Theorem 4.5. Indeed,

$\displaystyle\left\langle[p,q,u],v\right\rangle_{\mathbb{R}}$
$\displaystyle=$
$\displaystyle\left\langle(pq)u-p(qu),v\right\rangle_{\mathbb{R}}$
$\displaystyle=$
$\displaystyle\big{\langle}u,\overline{(pq)}v-\overline{q}(\overline{p}v)\big{\rangle}_{\mathbb{R}}$
$\displaystyle=$ $\displaystyle\left\langle
u,[\overline{q},\overline{p},v]\right\rangle_{\mathbb{R}}$ $\displaystyle=$
$\displaystyle\left\langle u,[q,p,v]\right\rangle_{\mathbb{R}}.$
$\displaystyle=$ $\displaystyle-\left\langle
u,[p,q,v]\right\rangle_{\mathbb{R}}.$

Due to identities (4.3) and (4.11) we obtain

$\displaystyle[p,u,v]$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{7}e_{i}\left<[e_{i},p,u],v\right>_{\mathbb{R}}$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{7}e_{i}\left<[e_{i},p,v],u\right>_{\mathbb{R}}$
$\displaystyle=$ $\displaystyle-[p,v,u].$

This proves (4.12).

∎

As a direct consequence of Lemma 4.6, we have for any $p\in\mathbb{O}$ and
$u\in H$

$[p,u,u]=0.$

The next lemma tells us how it works when factors of octonions pass through
the $\mathbb{O}$-inner product.

######  Lemma 4.7.

Let $H$ be a pre-Hilbert left $\mathbb{O}$-module. For any $u,v\in H$ and
$p,q\in\mathbb{O}$, the following identities hold:

(4.13) $\displaystyle\left<u,pv\right>$ $\displaystyle=$
$\displaystyle\left<u,v\right>\overline{p}+[p,u,v];$ (4.14)
$\displaystyle\left<pu,qv\right>$ $\displaystyle=$ $\displaystyle
p(\left<u,v\right>\overline{q})+p[q,u,v]+[p,u,qv];$ (4.15)
$\displaystyle\left<pu,qv\right>$ $\displaystyle=$
$\displaystyle(p\left<u,v\right>)\overline{q}+[pq,u,v]+\left<[p,q,v],u\right>.$

###### Proof.

We first prove (4.13). By the octonionic hermiticity, we conclude from (4.12)
that

$\displaystyle\left<u,pv\right>$ $\displaystyle=$
$\displaystyle\overline{\left<pv,u\right>}$ $\displaystyle=$
$\displaystyle\overline{p\left<v,u\right>+[p,v,u]}$ $\displaystyle=$
$\displaystyle\left<u,v\right>\overline{p}-{[p,v,u]}$ $\displaystyle=$
$\displaystyle\left<u,v\right>\overline{p}+[p,u,v].$

(4.14) can be deduced from (4.13) directly. Indeed,

$\displaystyle\left<pu,qv\right>$ $\displaystyle=$ $\displaystyle
p\left<u,qv\right>+[p,u,qv]$ $\displaystyle=$ $\displaystyle
p(\left<u,v\right>\overline{q})+p{[q,u,v]}+[p,u,qv].$

It follows from identities (4.4) and (4.12) that

$p[q,u,v]+[p,u,qv]=\left<[p,q,v],u\right>-[p,q,\left<v,u\right>]+[pq,u,v].$

Combining this with (4.14), we get

$\displaystyle\left<pu,qv\right>$ $\displaystyle=$ $\displaystyle
p(\left<u,v\right>\overline{q})+\left<[p,q,v],u\right>-[p,q,\left<v,u\right>]+[pq,u,v]$
$\displaystyle=$ $\displaystyle
p(\left<u,v\right>\overline{q})+[p,\left<u,v\right>,\overline{q}]+\left<[p,q,v],u\right>+[pq,u,v]$
$\displaystyle=$
$\displaystyle(p\left<u,v\right>)\overline{q}+[pq,u,v]+\left<[p,q,v],u\right>.$

This completes the proof. ∎

######  Remark 4.8.

All discussions in this subsection still hold when $\mathbb{O}$ is replaced by
an arbitrary real normed division algebra $\mathbb{K}$. This verifies that the
Hilbert space over any normed division algebras can be defined uniformly.

###  4.2. Norm of $\mathbb{O}$-Hilbert space

In this subsection, we collect some known results about the norm of an
$\mathbb{O}$-Hilbert space; see [3]. Our method is suitable for any Hilbert
space over any normed division algebras.

######  Definition 4.9.

A normed left $\mathbb{O}$-module is a pair $(X,\|\cdot\|)$ consisting of a
left $\mathbb{O}$-module $M$ and a map $\|\cdot\|:\ X\rightarrow\mathbb{R}$
satisfying the following axioms.

  1. (${\mathrm{i}}$)

$\left|\left|x\right|\right|\geqslant 0$ for all $x\in X$ with equality if and
only if $x=0$.

  2. (${\mathrm{ii}}$)

$\left|\left|px\right|\right|=|p|\left|\left|x\right|\right|$ for all
$p\in\mathbb{O}$ and $x\in X$.

  3. (${\mathrm{iii}}$)

$\left|\left|x+y\right|\right|\leqslant\left|\left|x\right|\right|+\left|\left|y\right|\right|$
for all $x,y\in X$.

A normed $\mathbb{O}$-module is said to be a Banach $\mathbb{O}$-module if it
is complete with respect to its canonical norm.

######  Definition 4.10.

An $\mathbb{O}$-Hilbert space is a pre-Hilbert left $\mathbb{O}$-module $H$
which is complete with respect to the norm
$\left|\left|\cdot\right|\right|:H\rightarrow\mathbb{R}^{+}$ induced by the
$\mathbb{O}$-inner product, i.e.,

(4.16) $\displaystyle\left|\left|u\right|\right|=\sqrt{\left<u,u\right>}.$

It is easy to check that in an $\mathbb{O}$-Hilbert space there holds the
polarization identity:

(4.17)
$\displaystyle\left<u,v\right>=\frac{1}{4}\sum_{i=0}^{7}e_{i}\left(\left|\left|e_{i}u+v\right|\right|^{2}-\left|\left|e_{i}u-v\right|\right|^{2}\right).$

This means that the $\mathbb{O}$-inner-product can be recovered from the norm.

######  Theorem 4.11.

Let $H$ be a Hilbert left $\mathbb{O}$-module. Then $H$ is a Banach left
$\mathbb{O}$-module with respect to the norm induced from the
$\mathbb{O}$-inner product.

###### Proof.

Since $(H,\left|\left|\cdot\right|\right|)$ is a real Banach space, it remains
to show that

$\left|\left|pu\right|\right|=\left|p\right|\left|\left|u\right|\right|$

for any $p\in\mathbb{O}$ and $x\in X$. According to (4.12) and (4.15) we
obtain

$\displaystyle\left<pu,pu\right>$
$\displaystyle=(p\left<u,u\right>)\overline{p}+[{pp},u,u]+\left<[p,p,u],u\right>=|p|^{2}\left|\left|u\right|\right|^{2}.$

This proves that $H$ is a Banach left $\mathbb{O}$-module. ∎

The Cauchy-Schwarz inequality has been established in [3]. Here we give an
alternative proof based on Lemma 4.7.

######  Proposition 4.12.

For all $u,v\in H$, we have the Cauchy-Schwarz inequality

(4.18)
$\displaystyle\left|\left<u,v\right>\right|\leqslant\left|\left|u\right|\right|\left|\left|v\right|\right|.$

Equality holds in (4.18) if and only if $u$ and $v$ are $\mathbb{O}$-linearly
dependent.

###### Proof.

To prove the Cauchy-Schwarz inequality, without loss of generality we may
assume $\left|\left|v\right|\right|=1$. Denote $p=\left<u,v\right>$. It
follows from (4.13) that

$\displaystyle\left<u-pv,pv\right>_{\mathbb{R}}$ $\displaystyle=$
$\displaystyle\text{Re}\,(\left<u-pv,v\right>\overline{p})$ $\displaystyle=$
$\displaystyle\text{Re}\,((\left<u,v\right>-p\left<v,v\right>)\ \overline{p})$
$\displaystyle=$ $\displaystyle 0.$

This implies that

(4.19)
$\displaystyle\left|\left|u\right|\right|^{2}=\left|\left|u-pv\right|\right|^{2}+\left|\left|pv\right|\right|^{2}\geqslant\left|\left|pv\right|\right|^{2}.$

Since

$\left|\left|pv\right|\right|=\left|p\right|\left|\left|v\right|\right|=\left|p\right|,$

we can rewrite (4.19) as

$\left|\left<u,v\right>\right|^{2}\leqslant\left|\left|u\right|\right|^{2}=\left|\left|u\right|\right|^{2}\left|\left|v\right|\right|^{2}.$

Moreover, in view of (4.19), we conclude that
$\left|\left<u,v\right>\right|=\left|\left|u\right|\right|\left|\left|v\right|\right|$
if and only if

$\left|\left|u-pv\right|\right|^{2}=0,$

which means $u,v$ are $\mathbb{O}$-linearly dependent. ∎

##  5\. Riesz representation theorem

Let $H$ be an $\mathbb{O}$-Hilbert space. We consider

$H^{*}:=\\{f\in{\text{Hom}_{L\mathbb{O}}(H,\mathbb{O})}\mid f\text{ is
continuous}\\}.$

For any $f\in H^{*}$, we define the right scalar multiplication by

(5.1) $\displaystyle(f\odot r)(x)\coloneqq\ f(x)r-[r,x,f]$

and define the norm

(5.2)
$\displaystyle\left|\left|f\right|\right|=\sup_{\left|\left|x\right|\right|\leqslant
1}\left|\left|f(x)\right|\right|.$

This makes $H^{*}$ a Banach right $\mathbb{O}$-module. Indeed, Theorem 3.10
shows that $H^{*}$ is a right $\mathbb{O}$ module. To show it is a normed
space, we need to prove

$\left|\left|f\odot p\right|\right|=\left|\left|f\right|\right|\
\left|p\right|$

for all $f\in H^{*}$ and $p\in\mathbb{O}$. This needs an alternative
expression of the right multiplication in (5.1) below.

######  Lemma 5.1.

For any $f\in H^{*}$ and $p\in\mathbb{O}$, we have

(5.3) ${(f\odot p)(x)}=pf(p^{-1}x)p.$

###### Proof.

Without loss of generality we can assume $|p|=1$. This means
$p^{-1}=\overline{p}$. By the definition of the second associator in (3.1), we
have

$\displaystyle f(\overline{p}x)=\overline{p}f(x)+[\overline{p},x,f].$

Multiplying the left and right above by $p$, we obtain

$pf(p^{-1}x)p=\Big{(}p\big{(}\overline{p}f(x)+[\overline{p},x,f]\big{)}\Big{)}p.$

We thus can apply the identity

$[\overline{p},x,f]=-[p,x,f]$

and identity (3.11)

$p[p,x,f]=[p,x,f]\overline{p}$

to conclude

(5.4) $\displaystyle pf(p^{-1}x)p=f(x)p-[p,x,f].$

By the definition of the right scalar multiplication in (3.14), the right side
above is exactly $(f\odot p)(x)$ as desired. ∎

######  Remark 5.2.

In our typical Example 3.3, we have $\mathbb{O}$-para-linear function
$R_{q}\in\text{Hom}_{L\mathbb{O}}(\mathbb{O},\mathbb{O})$ and its the right
multiplication by an octonion $p$ is given by

$R_{q}\odot p=R_{qp}.$

Indeed,

$\displaystyle R_{q}\odot p(x)$ $\displaystyle=$ $\displaystyle
R_{q}(x)p-[p,x,R_{q}]$ $\displaystyle=$ $\displaystyle(xq)p-[p,x,q]$
$\displaystyle=$ $\displaystyle(xq)p-[x,q,p]$ $\displaystyle=$ $\displaystyle
x(qp)$ $\displaystyle=$ $\displaystyle R_{qp}(x).$

In this setting, identity (5.3) becomes

$\displaystyle{(R_{q}\odot p)(x)}=pR_{q}(p^{-1}x)p.$

That is,

(5.5) $x(qp)=p((p^{-1}x)q)p$

for any $p,q,x\in\mathbb{O}$. This is a variant of the following Moufang
identity [18]

(5.6) $\displaystyle(xy)(ax)=x(ya)x.$

Consequently, (5.3) is a generalized Moufang identity.

######  Theorem 5.3.

$H^{*}$ is a Banach right $\mathbb{O}$-module.

###### Proof.

For any $f\in H^{*}$ and $p\in\mathbb{O}$, it follows from identity (5.3) that

$\displaystyle\left|\left|f\odot p\right|\right|$ $\displaystyle=$
$\displaystyle\sup_{\left|\left|x\right|\right|\leqslant
1}\left|\left|pf(p^{-1}x)p\right|\right|$ $\displaystyle\leqslant$
$\displaystyle\sup_{\left|\left|x\right|\right|\leqslant 1}\left|p\right|\
\left|\left|f\right|\right|\ \|p^{-1}x\|\ \left|p\right|$
$\displaystyle\leqslant$ $\displaystyle|p|\left|\left|f\right|\right|.$

Thus $f\odot p$ is a continuous $\mathbb{O}$-para-linear functional, i.e.,
$f\odot p\in H^{*}$. In view of Theorem 3.10, we conclude that $H^{*}$ is an
$\mathbb{O}$-submodule of the right $\mathbb{O}$-module
$\text{Hom}(H,\mathbb{O})$.

Since $p$ and $f$ are arbitrarily fixed, we can replace $p$ by $p^{-1}$ and
$f$ by $f\odot p$ to get

$\displaystyle\left|\left|(f\odot p)\odot
p^{-1}\right|\right|\leqslant|p^{-1}|\left|\left|f\odot p\right|\right|$

so that

$\displaystyle\left|\left|f\right|\right|$ $\displaystyle=$
$\displaystyle\left|\left|(f\odot p)\odot p^{-1}\right|\right|$
$\displaystyle\leqslant$ $\displaystyle|p^{-1}|\left|\left|f\odot
p\right|\right|$ $\displaystyle\leqslant$ $\displaystyle|p^{-1}|\ |p|\
\left|\left|f\right|\right|$ $\displaystyle=$
$\displaystyle\left|\left|f\right|\right|.$

This implies

$\left|\left|f\odot p\right|\right|=|p|\left|\left|f\right|\right|.$

Hence $H^{*}$ is a normed right $\mathbb{O}$-module with respect to norm
(5.2).

Finally, we show that $H^{*}$ is a Banach right $\mathbb{O}$-module. Let
$H^{*_{\mathbb{R}}}$ stand for the real dual space of $H$ regarded as a real
Hilbert space, i.e.,

$H^{*_{\mathbb{R}}}=\\{f:H\xlongrightarrow[]{}\mathbb{R}\ |\ \mbox{$f$ is a
bounded real linear functional}\\}.$

It is well-known that $H^{*_{\mathbb{R}}}$ is a real Banach space. Therefore,
if $f_{n}\in H^{*}$ is a Cauchy sequence, then there exists $f\in
H^{*_{\mathbb{R}}}$ such that $\lim_{n\to\infty}f_{n}=f$. It is easy to check
that the maps

$[p,x,\cdot]:H^{*}\to\mathbb{O}$

and

$\text{Re}\,:\mathbb{O}\to\mathbb{O}$

are both continuous. Hence

$\text{Re}\,[p,x,f]=\text{Re}\,([p,x,\lim_{n\to\infty}f_{n}])=\text{Re}\,(\lim_{n\to\infty}[p,x,f_{n}])=\lim_{n\to\infty}\text{Re}\,([p,x,f_{n}])=0.$

This proves $f\in H^{*}$. ∎

The following lemma shows that $H^{*}$ as a real Hilbert space is isometric
with $H^{*_{\mathbb{R}}}$ via the real part operator.

######  Lemma 5.4.

The real part map

$\displaystyle\text{Re}\,:H^{*}$ $\displaystyle\xlongrightarrow[]{}$
$\displaystyle H^{*_{\mathbb{R}}},$

defined by $(Ref)(x):=f_{\mathbb{R}}(x)$, is a norm-preserving map. That is,
for any $f\in H^{*}$ we have

$\left|\left|f\right|\right|_{H^{*}}=\left|\left|f_{\mathbb{R}}\right|\right|_{H^{*_{\mathbb{R}}}}.$

###### Proof.

Let $f\in H^{*}$ and $x\in H$ with $\left|\left|x\right|\right|\leqslant 1$.
Then there exists $p\in\mathbb{O}$ with $|p|=1$ such that

$|f(x)|=pf(x)=\text{Re}\,(pf(x)).$

Since $f$ is para-linear, it follows that

$|f(x)|=\text{Re}\,(pf(x))=\text{Re}\,(f(px))=f_{\mathbb{R}}(px)\leqslant\left|\left|f_{\mathbb{R}}\right|\right|_{H^{*_{\mathbb{R}}}}\left|\left|px\right|\right|\leqslant\left|\left|f_{\mathbb{R}}\right|\right|_{H^{*_{\mathbb{R}}}}.$

Thus we obtain
$\left|\left|f\right|\right|_{H^{*}}\leqslant\left|\left|f_{\mathbb{R}}\right|\right|_{H^{*_{\mathbb{R}}}}$.

On the other hand, it is easy to see that
$\left|\left|f_{\mathbb{R}}\right|\right|_{H^{*_{\mathbb{R}}}}\leqslant\left|\left|f\right|\right|_{H^{*}}$.
This completes the proof. ∎

To state the octonionic version of Riesz representation theorem, we need
introduce the notion of conjugate $\mathbb{O}$-linearity.

######  Definition 5.5.

Let $M_{1}$ be a left $\mathbb{O}$-module and $M_{2}$ a right
$\mathbb{O}$-module. A map $f\in\text{Hom}_{\mathbb{R}}(M_{1},M_{2})$ is
called conjugate $\mathbb{O}$-linear if

$f(px)=f(x)\overline{p}$

for all $p\in\mathbb{O}$ and all $x\in M_{1}$.

######  Theorem 5.6 (Riesz representation theorem).

Let $H$ be an $\mathbb{O}$-Hilbert space. Then there is an isometric conjugate
$\mathbb{O}$-isomorphism

(5.7) $\displaystyle H^{*}\cong{H}.$

###### Proof.

Let $f\in H^{*}$ be para-linear. By Theorem 3.4, it can be written as

$f(x)=f_{\mathbb{R}}(x)-\sum_{i=1}^{7}e_{i}f_{\mathbb{R}}(e_{i}x),$

where $f:H\to\mathbb{R}$ is a real linear functional. Then the classical Riesz
representation theorem shows there exists a unique element $z_{f}\in H$ such
that

$f_{\mathbb{R}}(x)=\left<x,z_{f}\right>_{\mathbb{R}}$

and

$\left|\left|z_{f}\right|\right|=\left|\left|f_{\mathbb{R}}\right|\right|.$

Since both $f$ and $\langle\cdot,z_{f}\rangle$ are $\mathbb{O}$-para-linear
and share the same real part, they must coincide as shown by Corollary 3.7,
i.e.,

$f(x)=\left<x,z_{f}\right>.$

By Lemma 5.4 we have

$\left|\left|z_{f}\right|\right|=\left|\left|f\right|\right|.$

Recall that $H^{*}$ is a Banach right $\mathbb{O}$-module by Theorem 5.3.
Hence, the map

$\displaystyle\sigma:H^{*}$ $\displaystyle\rightarrow$ $\displaystyle H$
$\displaystyle f$ $\displaystyle\mapsto$ $\displaystyle z_{f}$

is an isometry between $H^{*}$ and $H$ as real Banach spaces.

We next prove that $\sigma$ is a conjugate $\mathbb{O}$-homomorphism. For any
$x\in H$ and any $r\in\mathbb{O}$, by definition (3.14) we have

$\displaystyle\left<x,z_{f\odot r}\right>$ $\displaystyle=$
$\displaystyle{(f\odot r)(x)}$ $\displaystyle=$ $\displaystyle f(x)r-[r,x,f]$
$\displaystyle=$
$\displaystyle\left<x,z_{f}\right>\overline{\overline{r}}+[\overline{r},x,z_{f}]$
$\displaystyle=$ $\displaystyle\left<x,\overline{r}z_{f}\right>.$

This shows that

$\sigma{(f\odot r)}=\overline{r}\sigma(f).$

This implies that $\sigma$ is conjugate $\mathbb{O}$-linear.

We finally show that $\sigma$ is a conjugate $\mathbb{O}$-isomorphism. We
define

$\displaystyle\tau:H$ $\displaystyle\to$ $\displaystyle H^{*}$ $\displaystyle
y$ $\displaystyle\mapsto$ $\displaystyle y^{\prime}:=\left<\cdot,y\right>.$

We now prove $\tau$ is also a conjugate $\mathbb{O}$-homomorphism. For any
$x\in H$ and any $r\in\mathbb{O}$, it follows from (3.14) and (4.2) that

$\displaystyle{(y^{\prime}\odot r)(x)}$
$\displaystyle={y^{\prime}(x)}r-[r,x,y^{\prime}]$
$\displaystyle=\left<x,y\right>r-[r,x,y]$
$\displaystyle=\left<x,y\right>\overline{\overline{r}}+[\overline{r},x,y]$
$\displaystyle=\left<x,\overline{r}y\right>$
$\displaystyle={(\overline{r}y)^{\prime}(x)}.$

This shows that

$\tau(y)\odot r=\tau(\overline{r}y),$

i.e., $\tau$ is also a conjugate $\mathbb{O}$-homomorphism.

Moreover, it is easy to check that

$\sigma\tau=id,\quad\tau\sigma=id.$

Hence $\sigma$ is a conjugate $\mathbb{O}$-isomorphism. This finishes the
proof. ∎

Associated to a right $\mathbb{O}$-module $M$ is an induced left
$\mathbb{O}$-module $M^{-}$ with multiplication defined by

$p\cdot x:=x\overline{p}$

for all $x\in M$ and all $p\in\mathbb{O}$. We can thus identify two left
modules $H$ and $({H^{*}})^{-}$ by the isomorphism $\sigma$ in Theorem 5.6.
Moreover, this isomorphism induces on $({H^{*}})^{-}$ a canonical
$\mathbb{O}$-inner product

$\left<f,g\right>:={\left<z_{f},z_{g}\right>}$

for any $f,g\in(H^{*})^{-}$. By definition, the octonion hermiticity and
positivity can be verified directly. It remains to show the para-linearity. By
calculation, for any $r\in\mathbb{O}$ and $f,g\in H^{*}$ we have

$\displaystyle\left<r\cdot f,g\right>$ $\displaystyle=$
$\displaystyle\left<f\odot\overline{r},g\right>$ $\displaystyle=$
$\displaystyle\left<z_{f\odot\overline{r}},z_{g}\right>$ $\displaystyle=$
$\displaystyle\left<rz_{f},z_{g}\right>$ $\displaystyle=$ $\displaystyle
r\left<z_{f},z_{g}\right>+[r,z_{f},z_{g}].$

This shows that $\left<\cdot,g\right>$ is para-linear as desired. Then
$\sigma$ is an isomorphism between two Hilbert left $\mathbb{O}$-modules.

Based on Theorem 5.6, we can characterize the set

$H^{*_{\mathbb{O}}}=\\{f:H\xlongrightarrow[]{}\mathbb{O}\ |\ \mbox{$f$ is a
bounded $\mathbb{O}$-linear functional}\\}$

in terms of associative elements of $H$. To do this, we need a criterion for
an element $x\in H$ to be an associative element.

######  Lemma 5.7.

Let $H$ be a Hilbert left $\mathbb{O}$-module and $x\in H$. Then
$x\in\mathscr{A}(H)$ if and only if

(5.8) $[p,x,y]=0$

for all $p\in\mathbb{O}$ and $y\in H.$

###### Proof.

If $x\in\mathscr{A}(H)$, then Corollary 3.7 shows that

$[p,x,y]=[p,x,y^{\prime}]=0,$

where $y^{\prime}$ is defined as in the proof of Theorem 5.6.

Conversely, if $x\in H$ satisfies hypothesis (5.8), then identity (4.5)
implies that

$\left<[q,p,x],y\right>_{\mathbb{R}}=0$

for any $p,q\in\mathbb{O}$ and $y\in H$. Since
$\left(H,\left<\cdot,\cdot\right>_{\mathbb{R}}\right)$ is a real Hilbert
space, we thus conclude that

$[q,p,x]=0$

for all $p,q\in\mathbb{O}$, which means that $x\in\mathscr{A}(H)$. This
completes the proof. ∎

######  Theorem 5.8.

Let $H$ be an $\mathbb{O}$-Hilbert space. Then there is an isometric
isomorphism of real Banach spaces

(5.9) $\displaystyle H^{*_{\mathbb{O}}}\cong\mathscr{A}(H).$

###### Proof.

According to Theorem 5.6, it suffices to show that

$\tau|_{\mathscr{A}(H)}:\mathscr{A}(H)\to H^{*_{\mathbb{O}}}$

is surjective.

If $f\in H^{*_{\mathbb{O}}}$, then Theorem 5.6 ensures the existence of an
element $z_{f}\in H$ such that

$f(x)=\left<x,z_{f}\right>.$

Obviously,

$\tau(z_{f})=f.$

It remains to show that $z_{f}\in\mathscr{A}(H)$. Since $f$ is
$\mathbb{O}$-linear, it follows from identity (4.2) that for any
$p\in\mathbb{O}$,

$[p,z_{f},x]=-[p,x,z_{f}]=-[p,x,f]=-(f(px)-pf(x))=0.$

Then we conclude from Lemma 5.7 that $z_{f}\in\mathscr{A}(H)$. This completes
the proof. ∎

######  Remark 5.9.

Isomorphisms (5.7) and (5.9) indicate that the objects in octonionic
functional analysis are $\mathbb{O}$-para-linear maps rather than
$\mathbb{O}$-linear maps.

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##  Appendix A Interval Topology

Let $X$ be a compact topological space and $f:X\to\mathbb{R}$ be a continuous
function. Let $E_{f}$ be the epigraph of $f:X\to\mathbb{R}$,
$\phi:E_{f}\to\mathbb{R}$ be the projection map, and $\pi:E_{f}\to T_{f}X$ be
the Reeb quotient map, which identifies two points if they are in the same
connected component of a level set.

Define a partial order on $T_{f}X$ by letting $p\leq q$ if there exists $x$ in
$X$ and $r,s$ in $\mathbb{R}$ such that $f(x)\leq r\leq s$ and $p=\pi(x,r)$
and $q=\pi(x,s)$. The fact that this defines a poset follows from the
following lemma:

######  Lemma A.1.

Let $(x,r)$ and $(y,s)$ be points in $E_{f}$ such that $r\leq s$. The
following are equivalent:

  1. i)

$\pi(x,r)\leq\pi(y,s)$.

  2. ii)

$\pi(x,s)=\pi(y,s)$.

  3. iii)

$\pi(x,t)=\pi(y,t)$ for all $t>s$.

###### Proof.

$i)\implies ii):$ Let $z$ be a point in $X$ such that $\pi(z,r)=\pi(x,r)$ and
$\pi(z,s)=\pi(y,s)$. Let $C$ be the connected component of $\phi^{-1}(r)$
containing $(x,r)$ and $D$ be the connected component in $\phi^{-1}(s)$
containing $(y,s)$. Let $C^{\prime}$ be the subset of $E_{f}$ obtained by
shifting up the second coordinates of points in $C$ to $s$. As $C$ contains
$(z,r)$ and $D$ contains $(z,s)$, both $C^{\prime}$ and $D$ contains $(z,s)$.
Since $C^{\prime}$ is connected, it is contained in $D$. Hence
$\pi(x,s)=\pi(y,s)$.

$ii)\implies iii)$ Note that $\pi(x,s)=\pi(y,s)\leq\pi(y,t)$. Now, the result
follows from the argument in the previous part.

$iii)\implies i)$ Let $(s_{n})$ be a decreasing sequence converging to $s$.
Let $C_{n}\times\\{s_{n}\\}$ be the connected component of $\phi^{-1}(s_{n})$
containing both $(x,s_{n})$ and $(y,s_{n})$. Note that $(C_{n})$ is a
decreasing sequence of closed connected sets in $X$. Let $C$ be the
intersection of $C_{n}^{\prime}s$. Note that $C\times\\{s\\}$ is contained in
$E_{f}$ and $x,y$ are elements of $C$. Let us show that $C$ is connected. Let
$V$ and $W$ be open subsets in $X$ such that $C$ is contained in the disjoint
union of $U$ of $V$ and $W$. Assume $x$ is in $V$. Let $U_{n}$ be the
complement of $C_{n}$ in $X$. The collection consisting of $U$ and all
$U_{n}$’s is an open cover of $X$, hence it has a finite subcover. Since
$(U_{n})$ is an increasing family of open sets, this implies that for $n$
large enough $X$ is the union of $U$ and $U_{n}$, hence $C_{n}$ is contained
in $U$. Since $C_{n}$ is connected and has non-empty intersection with $V$ for
all $n$, for $n$ large enough $C_{n}$ is contained in $V$. Therefore $C$ is
contained in $V$. This shows $C$ is connected, hence $\pi(x,s)=\pi(y,s)$. By
definition, $\pi(x,r)\leq\pi(y,s).$ ∎

######  Corollary A.2.

  1. i)

$(T_{f}X,\leq)$ is a tree poset in the sense that the upper set of any point
is a chain (i.e., well ordered).

  2. ii)

Let $C$ be a chain in $(T_{f}X,\leq)$. Then $C$ has an infimum (i.e., the
maximum of its lower bounds), and the closure of $C$ in the quotient topology
is contained in the upper set of its infimum.

###### Proof.

$i):$ Assume that $\pi(x,r)\leq\pi(y,s)$, $\pi(x,r)\leq\pi(z,t)$ and $s\leq
t$. Then, by Lemma A.1, $\pi(y,t)=\pi(x,t)=\pi(z,t)$, and therefore
$\pi(y,s)\leq\pi(z,t)$.

$ii):$ let $(x_{n},r_{n})$ be a sequence in $C$ such that $\lim
r_{n}=r:=inf\\{s:(x,s)\in C\\}$. The sequence $(x_{n})$ has a subsequence
converging to a point $x$ in $X$. Let $(y_{n},s_{n})$ be a convergent sequence
in $E_{f}$ such that $\pi(y_{n},s_{n})$ is in $C$ for all $n$. Let
$lim(y_{n},s_{n})=(y,s)$. We need to show that $\pi(x,r)\leq\pi(y,s)$. Let
$t>s$. Without loss of generality, we can assume that $r_{n},s_{n}\leq t$ for
all $n$ and $(x_{n})$ converges to $x$. Since $C$ is a chain, by Lemma A.1,
$\pi(y_{i},t)=\pi(x_{j},t)$ for all $i,j$. Let $D$ be the connected component
of $\phi^{-1}(t)$ containing $(y_{n},t)$ and $(x_{n},t)$ for all $n$. Since
$D$ is closed, it contains $(x,t)$ and $(y,t)$, hence $\pi(x,t)=\pi(y,t)$.
Since $t>s$ was arbitrary, by Lemma A.1, $\pi(x,r)\leq\pi(y,s)$. This shows
$\pi(x,r)$ is a lower bound for the closure of $C$. Let
$\pi(x^{\prime},r^{\prime})$ be another lower bound for $C$. For any $t>r$,
there exists $n$ such that $r_{n}\leq t$, so
$\pi(x,t)=\pi(x_{n},t)=\pi(x^{\prime},t)$. Therefore, by Lemma A.1,
$\pi(x^{\prime},r^{\prime})\leq\pi(x,r)$. ∎

We introduce a new topology on $T_{f}X$, which we call the interval topology,
defined as follows.

######  Definition A.3.

Let $T_{f}X$ be the merge tree, viewed as a poset. A subset $U$ of $T_{f}X$ is
open in the interval topology if for each $p$ in $U$ and each $q$ in
$X\setminus\\{p\\}$ comparable to $p$, there exists $w$ in $U$ strictly in
between $p$ and $q$ such that the interval between $p$ and $w$ is contained in
$U$.

######  Proposition A.4.

If $(T_{f}X,\leq)$ has finitely many leaves (i.e., minimal elements in the
poset), then the quotient topology coincides with the interval topology.

######  Lemma A.5.

The upper set of any point is closed in the interval topology.

###### Proof.

Let $C$ be the upper set of a point $\pi(z,u)$ in $T_{f}X$. Let $U$ be its
complement. Let us show that $U$ is open in the interval topology. Let
$\pi(x,r)$ be in $U$. Let $\pi(y,s)$ be comparable to $\pi(x,r)$. If
$\pi(y,s)<\pi(x,r)$, then everything in between them is already in $U$ as $U$
is the complement of an upper set. Let us assume then $\pi(x,r)<\pi(y,s)$. By
Lemma A.1, there exists $t$ such that $r<t<s$ and $\pi(x,t)$ is not in the
upper set of $\pi(z,u)$. Then $\pi(x,t)$ is strictly in between $\pi(x,s)$ and
$\pi(y,s)$ and the interval in between $\pi(x,s)$ and $\pi(x,t)$ is in $U$. ∎

######  Lemma A.6.

The height map $h:T_{f}X\to\mathbb{R}$, $\pi(x,r)\mapsto r$ is continuous with
respect to the interval topology.

###### Proof.

Let $r$ be a real number and $\epsilon>0$. We need to show that the preimage
of $(r-\epsilon,r+\epsilon)$ is open in the interval topology. Assume that
$\pi(x,s)$ is in $T_{f}X$ and $|s-r|<\epsilon$. Let $\pi(y,t)$ be a point
comparable to $\pi(x,s)$. If $\pi(y,t)<\pi(x,s)$, then $\pi(x,s)=\pi(y,s)$.
Pick $s^{\prime}>t$ such that $r-\epsilon<s^{\prime}<s$. Then
$\pi(y,s^{\prime})$ is strictly in between $\pi(y,t)$ and $\pi(x,s)$ and the
image of the interval between $\pi(y,s^{\prime})$ and $\pi(x,s)$ is
$[s^{\prime},s]$, which is contained in $(r-\epsilon,r+\epsilon)$. If
$\pi(y,t)>\pi(x,s)$ , then $\pi(x,t)=\pi(y,t)$. Pick $s^{\prime}<t$ such that
$s<s^{\prime}<r+\epsilon$. Then $\pi(x,s^{\prime})$ is strictly in between
$\pi(x,s)$ and $\pi(y,t)$, and the image of the interval between $\pi(x,s)$
and $\pi(x,s^{\prime})$ is $[s.s^{\prime}]$, which is contained in
$(r-\epsilon,r+\epsilon)$. ∎

######  Lemma A.7.

For each $x$ in $X$, the map $\psi_{x}:[f(x),\infty)\to T_{f}X$,
$r\mapsto\pi(x,r)$ is continuous with respect to the interval topology.

###### Proof.

Let $U$ be an open set in the interval topology. Assume $\psi_{x}(r)=\pi(x,r)$
is in $U$. The result follows from the comparability of $\pi(x,r)$ with
$\pi(x,f(x))$ and $\pi(x,r+1)$. ∎

###### Proof of Proposition A.4.

Let $C_{1},\dots,C_{n}$ be the upper sets of leaves in $T_{f}X$. In both
topologies, they give a finite closed cover by Corollary A.2 and Lemma A.5.
This implies that a set $U$ is open in these topologies if and only if its
intersection with each $C_{i}$’s is relatively open. Hence, it is enough to
show that the relative topologies on the $C_{i}$’s coincide for the quotient
and the interval topologies. Both the height map $\pi(x,r)\mapsto r$ of Lemma
A.6 and $\psi_{x}$ of Lemma A.7 are continuous with respect to both
topologies. If we let $\pi(x,f(x))$ denote the leaf of $C_{i}$, and restrict
$h$ to $C_{i}$, then these maps becomes inverses of each other. This completes
the proof. ∎

We now return to the proof of Proposition 3.6 from the main text, which says
that the topology induced by the $\ell^{p}$-metrics on a merge tree agrees
with the quotient space topology.

###### Proof of Proposition 3.6.

Using the logic from the proof of Proposition A.4, it suffices to show that
the relative topologies on the $C_{i}$’s with respect to the interval and
metric topologies coincide. Indeed, the intersection of an $\ell^{p}$-ball
with each upper set $C_{j}$ of a leaf of the merge tree is relatively open
with respect to the interval topology. This is clear, since this intersection
is homeomorphic to an open interval via the height map. Similarly, relatively
open sets in the interval topology are relatively open in the metric topology.
∎

##  Appendix B Comparison and Existence of Merge Trees

In this paper we used two primary perspectives on merge trees: the classical
merge tree (Definition 2.6), which defines the merge tree as the Reeb graph of
the epigraph, and the generalized merge tree (Definition 2.5), which is
defined as the display poset of the persistent set. Each approach has its own
advantages, but in order to ensure that the Reeb graph of a function is
actually a graph requires specifying conditions on $f:X\to\mathbb{R}$ such as
the Morse condition or piecewise linearity [27, p. 857]. In this section, we
describe a third way of constructing the merge tree following the metric tree
construction in [50] and show that when $X$ is locally path connected, the
classical construction coincides with the metric construction described here.

For this section we assume that $X$ is a compact path-connected topological
space and $f:X\to\mathbb{R}$ is a continuous function. We are using the
definition of metric tree given in [8, Section 3.4], which is also called real
tree or $\mathbb{R}$-tree.

######  Remark B.1.

As the quotient of the epigraph of $f$, the merge tree is the wedge of the
image of the graph of $f$ and the half real line. Let us denote the image of
the graph of $f:X\to\mathbb{R}$ inside the merge tree by $M_{f}X$.

######  Definition B.2 (Metric Merge Tree).

Let $X$ be a compact path connected topological space and $f:X\to\mathbb{R}$
be a continuous function. Let us define $m_{f}:X\times X\to\mathbb{R}$ by

$m_{f}(x,y):=\inf\\{\max f\circ\gamma|\gamma:[a,b]\to
X,\gamma(a)=x,\gamma(b)=y\\}.$

Define $t_{f}:X\times X\to\mathbb{R}$ by

$t_{f}(x,y)=2m_{f}(x,y)-f(x)-f(y).$

It is easy to see that $t_{f}$ is a pseudo-metric (symmetric, non-negative and
satisfies triangle inequality). Let us denote the associated metric space by
$(\mathcal{T}_{f}X,t_{f})$ and call it metric merge tree. The function $f$ is
still well defined on $\mathcal{T}_{f}X$.

######  Proposition B.3.

Let $X$ be a compact topological space and $f:X\to\mathbb{R}$ be a continuous
function. Then $(\mathcal{T}_{f}X,t_{f})$ is a metric tree.

###### Proof.

It is enough to show that $t_{f}$ is a path metric and it has hyperbolicity
zero (see [8, Proposition 3.4.2]). To show that $t_{f}$ is a path metric, it
is enough to show that for each $x,y$ i $X$ and $\epsilon>0$, there exists $z$
in $X$ such that

$t_{f}(x,z)\leq\frac{t_{f}(x,y)}{2}+\epsilon,\qquad
t_{f}(y,z)\leq\frac{t_{f}(x,y)}{2}+\epsilon$

(see [11, Theorem 2.4.16]). Take $x,y$ in $X$ and $\epsilon>0$. Let
$\gamma:[0,1]\to X$ be a continuous curve from $x$ to $y$ such that

$m_{f}(x,y)\geq\max f\circ\gamma\epsilon.$

Let $M:=\max f\circ\gamma$. Define

$t_{0}:=\min\\{t:f(\gamma(t))=M\\},\qquad t_{1}:=\max\\{t:f(\gamma(t))=M\\}.$

Let $d_{0}:=M-f(x)$ and $d_{1}:=M-f(y)$. Without loss of generality
$d_{0}>=d_{1}$. Let $r:=(d_{0}+d_{1})/2$ and

$s:=\min\\{t:f\circ\gamma(t)=f(x)+r\\}.$

Let $z=\gamma(s)$. Now, we have:

$\displaystyle t_{f}(x,y)$ $\displaystyle\geq
2(M-\epsilon)-f(x)-f(y)=d_{0}+d_{1}-2\epsilon$ $\displaystyle t_{f}(x,z)$
$\displaystyle=r=\frac{d_{0}+d_{1}}{2}$ $\displaystyle t_{f}(y,z)$
$\displaystyle\leq(M-f(x)-r)+d_{1}=\frac{d_{0}+d_{1}}{2}.$

Now let us show that $t_{f}$ has $0$-hyperbolicity. Let $p$ be a point where
$f$ takes its maximum. The Gromov product $g_{p}:X\times X\to\mathbb{R}$ with
respect to the pseudo-metric $t_{f}$ is defined by:

$g_{p}(x,y)=\frac{1}{2}(t_{f}(p,x)+t_{f}(p,y)-t_{f}(x,y)).$

The metric $t_{f}$ has zero hyperbolicty if

$g_{p}(x,z)\geq\min(g_{p}(x,y),g_{p}(y,z))$

for all $x,y,z$ in $X$ (see [35, Corollary 1.1.B]).

Claim 1: $g_{p}(x,y)=f(p)-m_{f}(x,y)$ for all $x,y$ in $X$:

By maximality of $p$, $m_{f}(p,x)=f(p)$. Hence,

$\displaystyle t_{f}(p,x)=2m_{f}(p,x)-f(p)-f(x)=f(p)-f(x).$

So we have

$\displaystyle g_{p}(x,y)$
$\displaystyle=\frac{1}{2}(t_{f}(p,x)+t_{f}(p,y)-t_{f}(x,y))$
$\displaystyle=\frac{1}{2}(f(p)-f(x)+f(p)-f(y)-2m_{f}(x,y)+f(x)+f(y))$
$\displaystyle=f(p)-m_{f}(x,y)$

Claim 2: $g_{p}(x,z)\geq\min(g_{p}(x,y),g_{p}(y,z))$ for all $x,y,z$ in $X$.

By its definition, $m_{f}(x,z)\leq\max(m_{f}(x,y),m_{f}(y,z))$. Claim 2
follows from this and Claim 1.

Therefore the hyperbolicity of $\mathcal{T}_{f}X$ is $0$. ∎

######  Theorem B.4.

If $X$ is locally path connected, then $M_{f}X$ is the underlying topological
space of $\mathcal{T}_{f}X$.

###### Proof.

It is enough to show that $t_{f}(x,y)=0$ if and only if $(x,f(x))$ and
$(y,f(y))$ are identified under the Reeb quotient map $\pi$.

Assume $t_{f}(x,y)=0$. Then, $m_{f}(x,y)=f(x)=f(y)$. So, for each
$t>f(x)=f(y)$, there is a path in the sublevel set $X^{\leq t}$ connecting
$x,y$. Therefore, $\pi(x,t)=\pi(y,t)$, and by Lemma A.1, $\pi(x,r)=\pi(y,r)$.

Assume $\pi(x,f(x))=\pi(y,f(y))$. Let $r=f(x)$ and $t>r$. Since $x$ and $y$
are in the same connected component of the (closed) sublevel set $X^{\leq r}$
and $X$ is locally path connected, $x,y$ are in the same path component of the
(open) sublevel set $X^{<t}$. This implies that $m_{f}(x,y)<t$. Since $t>r$
arbitrary, $m_{f}(x,y)=r=f(x)=f(y)$, so $t_{f}(x,y)=0$.

∎

##  Appendix C Technical Proofs

This section contains proofs from the main body of the paper which are
technical, but essentially straightforward. These results are focused on
barcodes and bottleneck distances. To keep the exposition clean, we deal with
barcodes satisfying the simplifying assumptions:

  * •

the barcode is a _set_ (all multiplicities of intervals are equal to one);

  * •

the barcode is finite;

  * •

each interval in the barcode is a half open interval of the form $[b,d)$,
where $0\leq b\leq d\leq\infty$.

###  C.1. Proof of Proposition 3.16

To prove the proposition, we will introduce the notion of truncating a
barcode.

######  Definition C.1.

Let $B$ be a barcode and let $I=[b,d)\in B$. The truncation of $I$ at height
$h$, denoted $\mathrm{trunc}_{h}(I)$ is

  * •

equal to $I$ if $h\leq b$;

  * •

the interval $[h,d)$ if $b\leq h\leq d$;

  * •

the empty interval if $h>d$.

The truncation of $B$ at height $h$, $\mathrm{trunc}_{h}(B)$, is the barcode
obtained by truncating all of the intervals of $B$ at height $h$.

######  Lemma C.2.

Let $B$ and $B^{\prime}$ be barcodes whose intervals $[b,d)$ all satisfy
$b\geq H$ for some constant $H\geq 0$. Suppose that there exists a
$\delta$-matching between $B$ and $B^{\prime}$. For $\epsilon\leq\delta$,
there is a $\delta$-matching between $B$ and
$\mathrm{trunc}_{H+\epsilon}(B^{\prime})$.

###### Proof.

Let $\xi$ be a $\delta$-matching of $B$ and $B^{\prime}$. Define a matching
$\widehat{\xi}$ of $B$ and $\mathrm{trunc}_{H+\epsilon}(B^{\prime})$ by
setting

$\displaystyle\mathrm{dom}(\widehat{\xi})$
$\displaystyle:=\\{I\in\mathrm{dom}(\xi)\mid\mathrm{trunc}_{H+\epsilon}(\xi(I))\neq\emptyset\\}$
$\displaystyle\mathrm{ran}(\widehat{\xi})$
$\displaystyle:=\\{I^{\prime}\in\mathrm{ran}(\xi)\mid\mathrm{trunc}_{H+\epsilon}(I^{\prime})\neq\emptyset\\},\mbox{
and }$ $\displaystyle\widehat{\xi}(I)$
$\displaystyle:=\mathrm{trunc}_{H+\epsilon}(\xi(I)).$

Let $I=[b,d)\in\mathrm{dom}(\widehat{\xi})$ with
$I^{\prime}=[b^{\prime},d^{\prime})=\xi(I)$. Then $b^{\prime}\leq
H+\epsilon\leq d^{\prime}$ and
$\mathrm{trunc}_{H+\epsilon}(I^{\prime})=[H+\epsilon,d^{\prime})$. We have
$|d-d^{\prime}|\leq\delta$, by the assumption that $\xi$ was a
$\delta$-matching. Moreover,

$\delta\geq b-b^{\prime}\geq b-(H+\epsilon)$

and

$\delta\geq\epsilon\geq\epsilon+(H-b)=(H+\epsilon)-b$

imply that $|b-(H+\epsilon)|\leq\delta$, so that the cost of matching $I$ with
$\mathrm{trunc}_{H+\epsilon}(I^{\prime})$ is less than $\delta$. On the other
hand, if $I\not\in\mathrm{dom}(\widehat{\xi})$ then either
$I\not\in\mathrm{dom}(\xi)$, in which case we are done, or
$\mathrm{trunc}_{H+\epsilon}(\xi(I))=\emptyset$. In the latter case, let
$I=[b,d)$ and $I^{\prime}=[b^{\prime},d^{\prime})=\xi(I)$. We have

$d-b\leq d^{\prime}+\delta-b\leq d^{\prime}+\delta-H\leq
H+\epsilon+\delta-H\leq 2\delta,$

hence $\|[b,d)\|_{\Delta}\leq\delta$. Similar arguments handle intervals
$I^{\prime}\not\in\mathrm{ran}(\widehat{\xi})$. ∎

###### Proof of Proposition 3.16.

Let

$\widehat{d}_{B}(\mathcal{B}_{F},\mathcal{B}_{G}):=\inf\\{\epsilon\geq
0\mid\mbox{$\exists$ $\epsilon$-matching of $\mathcal{B}_{F}$ and
$\mathcal{B}_{G}$}\\},$

so that our goal is to show $d_{B}=\widehat{d}_{B}$. Clearly
$d_{B}\leq\widehat{d}_{B}$, since the former infimizes over a larger set of
matchings than the latter. To see the reverse inequality, let $\Phi$, $\Psi$
define an $(\epsilon,\delta)$-matching. If $\epsilon\geq\delta$, then each of
the $\delta$-matchings between $\mathcal{B}_{F}(p)$ and
$\mathcal{B}_{G}(\Phi(p))$ and between $\mathcal{B}_{G}(q)$ and
$\mathcal{B}_{F}(\Psi(q))$ is, in particular, an $\epsilon$-matching. This
implies the existence of an $\epsilon$-matching between $\mathcal{B}_{F}$ and
$\mathcal{B}_{G}$. Finally, suppose that $\delta\geq\epsilon$. Let
$\widehat{\Phi}$ and $\widehat{\Psi}$ denote the $\delta$-interleaving maps
obtained by composing $\Phi$ and $\Psi$ with flows in their respective target
merge trees. We claim that there exist $\delta$-matchings between all pairs
$\mathcal{B}_{F}(p)$ and $\mathcal{B}_{G}(\widehat{\Phi}(p))$ and between
$\mathcal{B}_{G}(q)$ and $\mathcal{B}_{F}(\widehat{\Psi}(q))$. Indeed, setting
$H=\pi_{F}(p)$, with $\pi_{F}:\mathcal{M}_{F}\to\mathbb{R}$ denoting
projection, we have that

$\mathcal{B}_{G}(\widehat{\Phi}(p))=\mathrm{trunc}_{H+\delta-\epsilon}(\mathcal{B}_{G}(\Phi(p))).$

Lemma C.2 therefore implies that there exists a $\delta$-matching of
$\mathcal{B}_{F}(p)$ and $\mathcal{B}_{G}(\widehat{\Phi}(p))$. The existence
of a $\delta$-matching of $\mathcal{B}_{G}(q)$ and
$\mathcal{B}_{F}(\widehat{\Psi}(q))$ follows for the same reason. ∎

We now have the tools at hand to prove Proposition 3.17.

###### Proof of Proposition 3.17..

Let $p\in\mathcal{M}_{F}$ be an arbitrary point. Choose a leaf $v$ of
$\mathcal{M}_{F}$ with $v\preccurlyeq p$ and a leaf $w\preccurlyeq\Phi(p)$.
Let $\pi_{F}:\mathcal{M}_{F}\to\mathbb{R}$ be the projection map. Then the
barcodes at these points can be expressed as

$\displaystyle\mathcal{B}_{F}(p)$
$\displaystyle=\mathrm{trunc}_{\pi_{F}(p)}(\mathcal{B}_{F}(v)),and$
$\displaystyle\mathcal{B}_{G}(\Phi(p))$
$\displaystyle=\mathrm{trunc}_{\pi_{F}(p)+\epsilon}(\mathcal{B}_{G}(w)).$

It is straightforward to check that a $\delta$-matching of
$\mathcal{B}_{F}(p)$ and $\mathcal{B}_{F}(\Phi(p))$ induces a
$\delta$-matching of the truncated barcodes. A similar argument applies to
pairs $q\in\mathcal{M}_{G}$ and $\Psi(q)\in\mathcal{M}_{F}$. ∎

###  C.2. Proof of Proposition 4.14

Let $B=\\{I_{j}=[b_{j},d_{j})\\}_{j=1}^{N}$ be the degree-$k$ barcode for $F$.
To ease exposition, assume that the births in the barcode are distinct—this
assumption is easily removed at the cost of necessitating more involved
notation. Suppose that the indices have been chosen so that
$b_{1}<b_{2}<\cdots<b_{N}$. For each bar $I_{j}$, we choose a representative
cycle $c_{j}$ which generates the persistent homology class represented by
$I_{j}$; in particular, choose $c_{j}$ so that all of its simplices are
contained in the connected component of the birth simplex for $I_{j}$. Let
$[c_{j}]$ denote the homology class of $c_{j}$. Let $p_{j}\in\mathcal{M}_{F}$
denote the birth point of $I_{j}$.

We can use these representative cycles to build bases for various homology
vector spaces which agree with the barcode decomposition. In the following,
let $\iota_{b,d}$ denote the inclusion map $F(b)\hookrightarrow F(d)$ for each
$b<d$. By an abuse of notation, we also let $\iota_{b,d}$ denote the induced
map on homology $H_{k}(F(b))\to H_{k}(F(d))$. We now construct our bases:

  1. (1)

$\\{v_{1}:=[c_{1}]\\}$ is a basis for $H_{k}(F(b_{1}))$,

  2. (2)

$\\{[c_{2}],\iota_{b_{1},b_{2}}(v_{1})\\}$ is a spanning set for
$H_{k}(F(b_{2}))$ and is linearly independent if

$\iota_{b_{1},b_{2}}(v_{1})\neq 0.$

Moreover, we can choose $\lambda_{1}^{(2)}\in\Bbbk$ so that

$v_{2}:=[c_{2}]+\lambda_{1}^{(2)}\iota_{b_{1},b_{2}}(v_{1})$

has the property that

$\iota_{b_{2},d}(v_{2})\not\in\mathrm{span}\\{\iota_{b_{1},d}(v_{1})\\}$

for any $d<d_{2}$. In particular, $v_{2}\not\in\mathrm{ker}(\iota_{b_{2},d})$
for $d<d_{2}$.

  3. (3)

Suppose that we have defined $v_{1},\ldots,v_{j-1}$. Then

$\\{[c_{j}],\iota_{b_{j-1},b_{j}}(v_{j-1}),\ldots,\iota_{b_{1},b_{j}}(v_{1})\\}$

is a spanning set for $H_{k}(F(b_{j}))$ which is linearly independent once any
zero vectors have been removed. We define

$v_{j}:=[c_{j}]+\lambda^{(j)}_{j-1}\iota_{b_{j-1},b_{j}}(v_{j-1})+\cdots+\lambda_{1}^{(j)}\iota_{b_{1},b_{j-1}}(v_{1}),$

where coefficients $\lambda^{(j)}_{i}$ are chosen so that

$\iota_{b_{j},d}(v_{j})\not\in\mathrm{span}\\{\iota_{b_{j-1},d}(v_{j-1}),\ldots,\iota_{b_{1},d}(v_{1})\\}$
(10)

for all $d<d_{j}$. In particular, $v_{j}\not\in\mathrm{ker}(\iota_{b_{j},d})$
for $d<d_{j}$.

The lifespan of the vector $v_{j}$ defined via this process represents the
persistent homology bar $I_{j}$.

To prove that $\widehat{\mathcal{F}}\approx\mathcal{F}$, we will show that
$\mathcal{F}$ is real interval decomposable. To do so, it suffices to show
that each of these $v_{j}$ can be chosen to be a linear combination of classes
with cycle representatives that lie in the same connected component as the
birth simplex of $I_{j}$. This will be achieved via induction.

For the base case of the induction argument, we have $v_{1}=[c_{1}]$, and the
claim follows immediately. Suppose that $v_{1},\ldots,v_{j-1}$ have the
desired property and consider $v_{j}$. We sort the $v_{1},\ldots,v_{j-1}$,
into a collection $v_{r_{1}},\ldots,v_{r_{m}}$ of vectors corresponding to
bars in $B$ which are born in the same component as $c_{j}$ (i.e., bars whose
birth points satisfy $p_{r_{1}},\ldots,p_{r_{m}}\preccurlyeq p_{j}$) and a
collection $v_{s_{1}},\ldots,v_{s_{n}}$ of vectors which do not have this
property, so that

$\displaystyle v_{j}$
$\displaystyle=[c_{j}]+\lambda_{r_{1}}^{(j)}\iota_{b_{r_{1}},b_{j}}(v_{r_{1}})+\cdots$
$\displaystyle\qquad+\lambda_{r_{m}}^{(j)}\iota_{b_{r_{m}},b_{j}}(v_{r_{m}})+\lambda_{s_{1}}^{(j)}\iota_{b_{s_{1}},b_{j}}(v_{s_{1}})+\cdots+\lambda_{s_{n}}^{(j)}\iota_{b_{s_{n}},b_{j}}(v_{s_{n}}).$

We claim that taking $\lambda_{s_{i}}^{(j)}=0$ for all $i=1,\ldots,n$ results
in a valid vector $v_{j}$; i.e., that this yields a $v_{j}$ satisfying (10).
Indeed, let $d<d_{j}$ and suppose that

$\displaystyle\iota_{b_{j},d}\left([c_{j}]+\lambda_{r_{1}}^{(j)}\iota_{b_{r_{1}},b_{j}}(v_{r_{1}})+\cdots+\lambda_{r_{m}}^{(j)}\iota_{b_{r_{m}},b_{j}}(v_{r_{m}})\right)$
$\displaystyle\hskip
144.54pt=\mu_{1}\iota_{b_{j},d}(v_{s_{1}})+\cdots+\mu_{m}\iota_{b_{j},d}(v_{s_{m}})$
(11)

for some coefficients $\mu_{i}$. We first note that there must be some
$v_{s_{i}}\not\in\mathrm{ker}(\iota_{s_{i},d})$—otherwise the right-hand side
of (11) is zero, hence left-hand side is as well and we arrive at the
contradiction that $\iota_{b_{j},d}(v_{j})=0$. We therefore have a well
defined number

$d^{\prime}:=\min\\{\mathrm{merge}(p_{j},p_{s_{i}})\mid\iota_{b_{s_{i}},d}(v_{s_{i}})\neq
0\\}.$

By the $H_{k}$-disjointness assumption, either $d_{j}<d^{\prime}$ or
$d_{s_{i}}<d^{\prime}$ for all $s_{i}$. The former case implies
$d<d^{\prime}$. By the induction hypothesis, there is a cycle representation
of the left-hand side of (11) whose vertices all belong to the same connected
component as those of $\iota_{b_{j},d}(c_{j})$ and a cycle representation of
the right-hand side whose vertices belong to a different connected component
than $c_{j}$. Moreover, these cycle representatives are homologous, and we
have arrived at a contradiction. On the other hand, if $d_{s_{i}}<d^{\prime}$
for all $s_{i}$ then it must be that $d<d_{s_{i}}$ for some $s_{i}$ (since
there exists some $v_{s_{i}}\not\in\mathrm{ker}(\iota_{b_{s_{i}},d})$) and a
similar argument yields a contradiction. This completes the proof. ∎