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\section*{Banach space}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{banach_spaces}{}\section*{{Banach spaces}}\label{banach_spaces}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{banach_spaces_as_metric_spaces}{Banach spaces as metric spaces}\dotfill \pageref*{banach_spaces_as_metric_spaces} \linebreak
\noindent\hyperlink{morphisms}{Maps between Banach spaces}\dotfill \pageref*{morphisms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{operations_on_banach_spaces}{Operations on Banach spaces}\dotfill \pageref*{operations_on_banach_spaces} \linebreak
\noindent\hyperlink{integration_in_banach_spaces}{Integration in Banach spaces}\dotfill \pageref*{integration_in_banach_spaces} \linebreak
\noindent\hyperlink{bochner_integral}{Bochner integral}\dotfill \pageref*{bochner_integral} \linebreak
\noindent\hyperlink{the_spectral_integral}{The spectral integral}\dotfill \pageref*{the_spectral_integral} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_bornological_spaces}{Relation to bornological spaces}\dotfill \pageref*{relation_to_bornological_spaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A Banach space $\mathcal{B}$ is both a [[vector space]] (over a [[normed field]] such as $\mathbb{R}$) and a [[complete space|complete]] [[metric space]], in a compatible way. Hence a complete [[normed vector space]].

A source of simple Banach spaces comes from considering a [[Cartesian space]] $\mathbb{R}^n$ (or $K^n$ where $K$ is the normed field) with the norm:

\begin{displaymath}
{\|(x_1,\ldots,x_n)\|_p} \coloneqq \root p {\sum_{i = 1}^n {|x_i|^p}}
\end{displaymath}
where $1 \leq p \leq \infty$ (this doesn't strictly make sense for $p = \infty$, but taking the limit as $p \to \infty$ and reading $\mathbb{R}^\infty = \underset{\longrightarrow}{\lim}_n \mathbb{R}^n$ as the [[direct limit]] (as opposed to the [[inverse limit]]) we arrive at the formula ${\|(x_1,\ldots,x_n)\|_\infty} \coloneqq \max_i {|x_i|}$).

However, the theory of these spaces is not much more complicated than that of finite-dimensional vector spaces because they all have the same underlying topology. When we look at infinite-dimensional examples, however, things become trickier. Common examples are [[Lebesgue spaces]], [[Hilbert spaces]], and [[sequence spaces]].

In the literature, one most often sees Banach spaces over the field $\mathbb{R}$ of [[real numbers]]; Banach spaces over the field $\mathbb{C}$ of [[complex numbers]] are not much different, since they are also over $\mathbb{R}$. But people do study them over [[p-adic numbers]] too. \emph{Unless otherwise stated, we assume $\mathbb{R}$ below.}

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $V$ be a [[vector space]] over the field of [[real number]]s. (One can generalise the choice of [[field]] somewhat.) A \textbf{pseudonorm} (or \textbf{[[seminorm]]}) on $V$ is a function

\begin{displaymath}
{\| - \|}\colon V \to \mathbb{R}
\end{displaymath}
such that:

\begin{enumerate}%
\item ${\|0\|} \leq 0$;
\item ${\|r v\|} = {|r|} {\|v\|}$ (for $r$ a [[scalar]] and $v$ a vector);
\item ${\|v + w\|} \leq {\|v\|} + {\|w\|}$.

\end{enumerate}
It follows from the above that ${\|v\|} \geq 0$; in particular, ${\|0\|} = 0$. A \textbf{[[norm]]} is a pseudonorm that satisfies a converse to this: $v = 0$ if ${\|v\|} = 0$.

A [[norm]] on $V$ is \textbf{complete} if, given any infinite [[sequence]] $(v_1, v_2, \ldots)$ such that

\begin{equation}
\lim_{m,n\to\infty} {\left\| \sum_{i=m}^{m+n} v_i \right\|} = 0 ,
\label{Cauchy}\end{equation}
there exists a (necessarily unique) \textbf{sum} $S$ such that

\begin{equation}
\lim_{n\to\infty} {\left\| S - \sum_{i=1}^n v_i \right\|} = 0 ;
\label{converge}\end{equation}
we write

\begin{displaymath}
S = \sum_{i=1}^\infty v_i
\end{displaymath}
(with the right-hand side undefined if no such sum exists).

Then a \textbf{Banach space} is simply a vector space equipped with a complete norm. As in the real line, we have in a Banach space that

\begin{displaymath}
{\left\| \sum_{i=1}^\infty v_i \right\|} \leq \sum_{i=1}^\infty {\|v_i\|} ,
\end{displaymath}
with the left-hand side guaranteed to exist if the right-hand side exists as a finite real number (but the left-hand side may exist even if the right-hand side diverges, the usual distinction between absolute and conditional convergence).

If we do not insist on the space being complete, we call it a \textbf{normed (vector) space}. If we have a [[topological vector space]] such that the topology comes from a norm, but we do not make an actual choice of such a norm, then we talk of a \textbf{normable space}.

\hypertarget{banach_spaces_as_metric_spaces}{}\subsubsection*{{Banach spaces as metric spaces}}\label{banach_spaces_as_metric_spaces}

The three axioms for a pseudonorm are very similar to the three axioms for a [[pseudometric]].

Indeed, in any pseudonormed vector space, let the \textbf{distance} $d(v,w)$ be

\begin{displaymath}
d(v,w) = {\|w - v\|} .
\end{displaymath}
Then $d$ is a pseudometric, which is \textbf{translation-invariant} in that

\begin{displaymath}
d(v+x,w+x) = d(v,w)
\end{displaymath}
always holds. Conversely, given any translation-invariant pseudometric $d$ on a vector space $V$, let ${\|v\|}$ be

\begin{displaymath}
{\|v\|} = d(0,v) .
\end{displaymath}
Then ${\|-\|}$ satisfies the axioms (1--3) for a pseudonorm, except that it may satisfy (2) only for $r = 0, \pm 1$. (In other words, it is only a [[G-pseudonorm]].) It will actually be a pseudonorm iff the pseudometric satisfies a homogeneity rule:

\begin{displaymath}
d(r v,r w) = {|r|} d(v,w) .
\end{displaymath}
Thus pseudonorms correspond precisely to homogeneous translation-invariant pseudometrics.

Similarly, norms correspond to homogenous translation-invariant metrics and complete norms correspond to complete homogeneous translation-invariant metrics. Indeed, \eqref{Cauchy} says that the sequence of partial sums is a [[Cauchy sequence]], while \eqref{converge} says that the sequence of partial sums converges to $S$.

Thus a Banach space may equivalently be defined as a vector space equipped with a complete homogeneous translation-invariant metric. Actually, one usually sees a sort of hybrid approach: a Banach space is a normed vector space whose corresponding metric is complete.

\hypertarget{morphisms}{}\subsubsection*{{Maps between Banach spaces}}\label{morphisms}

If $V$ and $W$ are pseudonormed vector spaces, then the \textbf{norm} of a linear function $f\colon V \to W$ may be defined in either of these equivalent ways:

\begin{itemize}%
\item ${\|f\|} = \sup \{ {\|f v\|} \;|\; {\|v\|} \leq 1 \}$;
\item ${\|f\|} = \inf \{ r \;|\; \forall{v},\; {\|f v\|} \leq r {\|v\|} \}$.

\end{itemize}
(Some other forms are sometimes seen, but these may break down in degenerate cases.)

For finite-dimensional spaces, any linear map has a well-defined finite norm. In general, the following are equivalent:

\begin{itemize}%
\item $f$ is [[continuous map|continuous]] (as measured by the pseudometrics on $V$ and $W$) at $0$;
\item $f$ is continuous (everywhere);
\item $f$ is [[uniformly continuous map|uniformly continuous]];
\item $f$ is [[Lipschitz map|Lipschitz continuous]];
\item ${\|f\|}$ is finite (and, in [[constructive mathematics]], [[located real number|located]]);
\item $f$ is [[bounded map|bounded]] (as measured by the [[bornologies]] given by the pseudometrics on $V$ and $W$).

\end{itemize}
In this case, we say that $f$ is \textbf{bounded}. If $f\colon V \to W$ is not assumed to be linear, then the above conditions are no longer equivalent.

The bounded linear maps from $V$ to $W$ themselves form a pseudonormed vector space $\mathcal{B}(V,W)$. This will be a Banach space if (and, except for degenerate cases of $V$, only if) $W$ is a Banach space. In this way, the category $Ban$ of Banach spaces is a [[closed category]] with $\mathbb{R}$ as the unit.

The clever reader will note that we have not yet defined $\mathbf{Ban}$ as a category! (surprisingly in the \emph{nLab}) There are many (nonequivalent) ways to do so.

In [[functional analysis]], the usual notion of `[[isomorphism]]' for Banach spaces is a bounded bijective linear map $f\colon V \to W$ such that the [[inverse function]] $f^{-1}\colon W \to V$ (which is necessarily linear) is also bounded. In this case one can accept all bounded linear maps between Banach spaces as morphisms. Analysts sometimes refer to this as the ``isomorphic category''.

Another natural notion of isomorphism is a surjective linear isometry. In this case, we take a morphism to be a \textbf{[[short map|short]]} [[linear map]], or [[short linear map|linear contraction]]: a linear map $f$ such that ${\|f\|} \leq 1$. This category, which is what category theorists generally refer to as $\mathbf{Ban}$, is sometimes referred to as the ``isometric category'' by analysts. Note that this makes the `underlying set' (in the sense of $\mathbf{Ban}$ as a [[concrete category]] like any closed category) of a Banach space its (closed) \textbf{unit ball}

\begin{displaymath}
Hom_Ban(\mathbb{R},V) \cong \{ v \;|\; {\|v\|} \leq 1 \}
\end{displaymath}
rather than the set of all vectors in $V$ (the underlying set of $V$ as a vector space).

Yemon Choi: This is really here to remind myself how to make query boxes. But while I'm at it, is it really OK to refer to the ``unit ball functor'' as ``taking the underlying set''? I notice that on the discussion about internal homs at [[internal hom]] it is claimed that ``Every closed category is a [[concrete category]] (represented by $I$), and the underlying set of the internal hom is the external hom'' which seems to require ``underlying set'' to be interpreted in this looser sense.

\emph{Toby}: Sure, but the point of putting `underlying set' in scare quotes is precisely to point out that the category-theoretic underlying set is not what one would normally expect.

Mark Meckes: I've expanded this section in part to be consistent with analysts' terminology. I've made some assumptions about category theorists' conventions which might not be correct. (If I find time I might write about other categories of Banach spaces that analysts think about.)

\emph{Toby}: Looks good to me!

From a category-theorist's perspective, the isomorphic category is really the [[full image]] of the [[inclusion functor]] from $Ban$ to $TVS$ (the category of [[topological vector spaces]]), which may be denoted $Ban_{TVS}$. If you're working in $Ban_{TVS}$, then you only care about the topological linear structure of your space (although you do also care that it can be derived from some metric); if you're working in $Ban$, then you care about all of the structure on the space.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Many examples of Banach spaces are parametrised by an exponent $1 \leq p \leq \infty$. (Sometimes one can also try $0 \leq p \lt 1$, but these generally don't give Banach spaces.)

\begin{itemize}%
\item The [[Cartesian space]] $\mathbb{R}^n$ is a Banach space with

\begin{displaymath}
{\|(x_1,\ldots,x_n)\|_p} = \root p {\sum_i {|x_i|^p}} .
\end{displaymath}
(We can allow $p = \infty$ by taking a limit; the result is that ${\|x\|_\infty} = \max_i {|x_i|}$.) Every finite-dimensional Banach space is isomorphic to this for some $n$ and $p$; in fact, once you fix $n$, the value of $p$ is irrelevant up to isomorphism.


\item The [[sequence space]] $l^p$ is the set of infinite [[sequence]]s $(x_1,x_2,\ldots)$ of real numbers such that

\begin{displaymath}
{\|(x_1,x_2,\ldots)\|_p} = \root p {\sum_i {|x_i|^p}}
\end{displaymath}
exists as a finite real number. (The only question is whether the sum converges. Again $p = \infty$ is a limit, with the result that ${\|x\|_\infty} = \sup_i {|x_i|}$.) Then $l^p$ is a Banach space with that norm. These are all versions of $\mathbb{R}^\infty$, but they are no longer isomorphic for different values of $p$. (See [[isomorphism classes of Banach spaces]].)


\item More generally, let $A$ be any [[set]] and let $l^p(A)$ be the set of [[function]]s $f$ from $A$ to $\mathbb{R}$ such that

\begin{displaymath}
{\|f\|_p} = \root p {\sum_{x: A} {|f(x)|^p}}
\end{displaymath}
exists as a finite real number. (Again, ${\|f\|_\infty} = \sup_{x\colon A} {|f(x)|}$.) Then $l^p(A)$ is a Banach space. (This example includes the previous examples, for $A$ a countable set.)


\item On any [[measure space]] $X$, the [[Lebesgue space]] $\mathcal{L}^p(X)$ is the set of measurable almost-everywhere-defined real-valued functions on $X$ such that

\begin{displaymath}
{\|f\|_p} = \root p {\int {|f|^p}}
\end{displaymath}
exists as a finite real number. (Again, the only question is whether the integral converges. And again $p = \infty$ is a limit, with the result that ${\|f\|_\infty}$ is the [[essential supremum]] of ${|f|}$.) As such, $\mathcal{L}^p(X)$ is a complete pseudonormed vector space; but we identify functions that are equal almost everywhere to make it into a Banach space. (This example includes the previous examples, for $X$ a set with counting measure.)


\item Any [[Hilbert space]] is Banach space; this includes all of the above examples for $p = 2$.



\end{itemize}
\hypertarget{operations_on_banach_spaces}{}\subsection*{{Operations on Banach spaces}}\label{operations_on_banach_spaces}

The category $Ban$ of Banach spaces is [[complete category|small complete]], [[cocomplete category|small cocomplete]], and [[symmetric monoidal closed category|symmetric monoidal closed]] with respect to its standard internal hom (described at [[internal hom]]). Some details follow.

\begin{itemize}%
\item The category of Banach spaces admits small [[product]]s. Given a small family of Banach spaces $\{X_\alpha\}_{\alpha \in A}$, its product in $Ban$ is the subspace of the vector-space product

\begin{displaymath}
\prod_{\alpha \in A} X_\alpha
\end{displaymath}
consisting of $A$-tuples $\langle x_\alpha \rangle$ which are \emph{uniformly} bounded (i.e., there exists $C$ such that $\forall \alpha \in A: {\|x_\alpha\|} \leq C$), taking the least such upper bound as the norm of $\langle x_\alpha \rangle$. This norm is called the $\infty$-norm; in particular, the product of an $A$-indexed family of copies of $\mathbb{R}$ or $\mathbb{C}$ is what is normally denoted as $l^{\infty}(A)$.


\item The category of Banach spaces admits [[equalizer]]s. Indeed, the equalizer of a pair of maps $f, g: X \rightrightarrows Y$ in $Ban$ is the [[kernel]] of $f-g$ under the norm inherited from $X$ (the kernel is closed since $f-g$ is continuous, and is therefore complete). In fact every equalizer is even a [[section]] by the [[Hahn-Banach theorem]]. Every [[extremal monomorphism]] is even already an equalizer (and a section): Let $f\colon X \to Y$ be an extremal monomorphism, $\iota\colon \Im(f) \to Y$ the embedding of $Im(f)$ into the codomain of $f$ and $f\prime \colon X \to Im(f)$ $f$ with restricted codomain. Since $f\prime$ is an epimorphism, $f=\iota f\prime$, and $f$ extremal, $f\prime$ is an isomorphism, thus $f$ is an embedding.


\item The category of Banach spaces admits small [[coproduct]]s. Given a small family of Banach spaces $\{X_\alpha\}_{\alpha \in A}$, its coproduct in $Ban$ is the completion of the vector space coproduct

\begin{displaymath}
\bigoplus_{\alpha \in A} X_\alpha
\end{displaymath}
with respect to the norm given by

\begin{displaymath}
{\left\| \bigoplus_{s \in S} x_s \right\|} = \sum_{s \in S} {\|x_s\|} ,
\end{displaymath}
where $S \subseteq A$ is finite and ${\|x_s\|}$ denotes the norm of an element in $X_s$. This norm is called the $1$-norm; in particular, the coproduct of an $A$-indexed family of copies of $\mathbb{R}$ or $\mathbb{C}$ is what is normally denoted as $l^1(A)$.


\item The category of Banach spaces admits [[coequalizer]]s. Indeed, the coequalizer of a pair of maps $f, g: X \rightrightarrows Y$ is the [[cokernel]] of $f-g$ under the quotient norm (in which the norm of a coset $y + C$ is the minimum norm attained by elements of $y + C$; here $C$ is the [[image]] $(f-g)(X)$, which is closed). It is standard that the quotient norm on $Y/C$ is complete given that the norm on $Y$ is complete.


\item To describe the tensor product $X \otimes_{Ban} Y$ of two Banach spaces (making $Ban$ symmetric monoidal closed with respect to its usual internal hom), let $F(X \times Y)$ be the free vector space generated by the set $X \times Y$, with norm on a typical element defined by

\begin{displaymath}
{\left\| \sum_{1 \leq i \leq n} a_i (x_i \otimes y_i) \right\|} = \sum_{1 \leq i \leq n} {|a_i|} {\|x_i\|} \cdot {\|y_i\|}.
\end{displaymath}
Let $\overline{F}(X \times Y)$ denote its completion with respect to this norm. Then take the cokernel of $\overline{F}(X \times Y)$ by the closure of the subspace spanned by the obvious bilinear relations. This quotient is $X \otimes_{Ban} Y$.



\end{itemize}
In the literature on Banach spaces, tensor product above is usually called the \textbf{projective tensor product} of Banach spaces; see other [[tensor product of Banach spaces]]. The product and coproduct are considered \textbf{direct sums}; see other [[direct sums of Banach spaces]].

To be described:

\begin{itemize}%
\item duals ($p + q = p q$);
\item completion ($Ban$ is a [[reflective subcategory]] of $PsNVect$ (pseudo-normed vector spaces)).
\item $Ban$ as a (somewhat larger) category with duals.

\end{itemize}
\hypertarget{integration_in_banach_spaces}{}\subsection*{{Integration in Banach spaces}}\label{integration_in_banach_spaces}

This paragraph describes some aspects of integration theory in Banach spaces that are relevant to understand the literature about [[AQFT]]. In the given context, elements of a Banach space $\mathcal{B}$ are sometimes called vectors, a function or measure taking values in $\mathcal{B}$ are therefore called vector functions and vector measures. Functions and measures taking values in the [[field]] that the Banach space is defined upon as a vector space are called scalar functions and scalar measures.

We will consider two types of integrals:

\begin{itemize}%
\item integrals of vector functions with respect to a scalar measure, specifically the Bochner integral,


\item integrals of scalar functions with respect to a vector measure, specifically the spectral integral of a normal operator on a Hilbert space.



\end{itemize}
\hypertarget{bochner_integral}{}\subsubsection*{{Bochner integral}}\label{bochner_integral}

The Bochner integral is a direct generalization of the Lesbegue integral to functions that take values in a Banach space. Whenever you encounter an integral of a function taking values in a Banach space in the [[AQFT]] literature, it is safe to assume that it is meant to be a Bochner integral. Two points already explained by Wikipedia are of interest:

\begin{enumerate}%
\item A version of the dominated convergence theorem is true for the Bochner integral.
\item There are theorems that are not valid for the Bochner integral, notably the Radon-Nikodym theorem does not hold in general.

\end{enumerate}
\begin{itemize}%
\item \href{http://en.wikipedia.org/wiki/Bochner_integral}{Wikipedia}

\end{itemize}
\emph{reference}: Joseph Diestel: ``Sequences and Series in Banach Spaces'' (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0542.46007&format=complete}{ZMATH entry}), chapter IV.

\hypertarget{the_spectral_integral}{}\subsubsection*{{The spectral integral}}\label{the_spectral_integral}

The integral with respect to the spectral measure of a bounded normal operator on a Hilbert space is an example of a Banach space integral with respect to a vector measure. In this paragraph we present a well known, but somewhat less often cited result, that is of use in some proofs in some approaches to [[AQFT]], it is the version of the dominated convergence theorem for the given setting.

Let A be a bounded normal operator on a Hilbert space and E be it's spectral measure (the ``resolution of identity'' in the terms of Dunford and Schwartz). Let $\sigma(A)$ be the spectrum of A. For a bounded complex Borel function f we then have

\begin{displaymath}
f(A) \coloneqq \int_{\sigma(A)} f(\lambda) E(d\lambda)
\end{displaymath}
\begin{utheorem}
If the uniformly bounded sequence $\{f_n\}$ of complex Borel functions converges at each point of $\sigma(A)$ to the function $f$, then $f_n(A) \to f(A)$ in the strong operator topology.

\end{utheorem}
\begin{proof}
\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_bornological_spaces}{}\subsubsection*{{Relation to bornological spaces}}\label{relation_to_bornological_spaces}

Every [[inductive limit]] of [[Banach spaces]] is a [[bornological vector space]]. (\hyperlink{AlpaySalomon13}{Alpay-Salomon 13, prop. 2.3})

Conversely, every [[bornological vector space]] is an inductive limit of [[normed spaces]], and of [[Banach spaces]] if it is quasi-complete (\hyperlink{SchaeferWolff99}{Schaefer-Wolff 99})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[reflexive Banach space]]


\item [[projective Banach space]]


\item [[Banach analytic space]]



\end{itemize}
[[!include analytic geometry ingredients -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after [[Stefan Banach]].

\begin{itemize}%
\item Walter Rudin, \emph{Functional analysis}


\item Dunford, Nelson; Schwartz, Jacob T.: ``Linear operators. Part I: General theory.'' (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0635.47001&format=complete}{ZMATH entry}), ``Linear operators. Part II: Spectral theory, self adjoint operators in Hilbert space.'' (\href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:0635.47002&format=complete}{ZMATH entry})


\item Z. Semadeni, \emph{Banach spaces of continuous functions}, vol. I, Polish scientific publishers. Warszawa 1971


\item Daniel Alpay, Guy Salomon, \emph{On algebras which are inductive limits of Banach spaces} (\href{http://arxiv.org/abs/1302.3372}{arXiv:1302.3372})


\item H. H. Schaefer with M. P. Wolff, \emph{Topological vector spaces}, Springer 1999



\end{itemize}
category: analysis

[[!redirects Banach space]] [[!redirects Banach spaces]] [[!redirects Banach vector space]] [[!redirects Banach vector spaces]]

[[!redirects Ban]]

[[!redirects norm isomorphism]] [[!redirects norm-isomorphic]] [[!redirects norm isomorphic]]



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