nLab
Banach space

Idea

A Banach space is both a vector space and a metric space, in a compatible way. In finite dimensions, every $n$-dimensional real Banach space may be described (up to linear isometry, the usual sort of isomorphism) as the Cartesian space ${ℝ}^n$ equipped with the $p$-norm for $1\le p\le \mathrm{\infty }$:

$$\parallel (x_1,\dots ,x_n){\parallel }_p≔\sqrt[p]{\sum _{i=1}^n\mid x_i{\mid }^p}$$\|(x_1,\ldots,x_n)\|_p \coloneqq \root p {\sum_{i = 1}^n |x_i|^p}

(or $\parallel (x_1,\dots ,x_n){\parallel }_{\mathrm{\infty }}≔{\max }_i\mid x_i\mid$ for $p=\mathrm{\infty }$).

This is a familiar space to most people, and even the unusual norms (the usual Euclidean norm corresponds to $p=2$) are easy to visualise. When we look at infinite-dimensional examples, however, things become trickier. Common examples may be drawn from measure theory, Hilbert spaces, and spaces of sequences.

Yemon, I don't want you to think that I'm trying to pull rank (as an established Lab contributor) on you as I edit what you've edited here. So please do edit further (or make a comment like this one) as you like. —Toby Bartels

Definitions

Let $V$ be a vector space over the field of real numbers. (One can generalise the choice of field somewhat.) A pseudonorm (or seminorm) on $V$ is a function

$$\parallel -\parallel :V\to ℝ$$\| - \|: V \to \mathbb{R}

such that:

1. $\parallel 0\parallel \le 0$;
2. $\parallel rv\parallel =\mid r\mid \parallel v\parallel$ (for $r$ a scalar and $v$ a vector);
3. $\parallel v+w\parallel \le \parallel v\parallel +\parallel w\parallel$.

It follows from the above that $\parallel v\parallel \ge 0$; in particular, $\parallel 0\parallel =0$. A norm is a pseudonorm that satisfies a converse to this: $v=0$ if $\parallel v\parallel =0$.

A norm on $V$ is complete if, given any infinite sequence $(v_1,v_2,\dots )$ such that

(1)$$\underset{m,n\to \mathrm{\infty }}{\lim }\parallel \sum _{i=m}^{m+n}{v}_i\parallel =0,$$\lim_{m,n\to\infty} \left\| \sum_{i=m}^{m+n} v_i \right\| = 0 ,

there exists a (necessarily unique) sum $S$ such that

(2)$$\underset{n\to \mathrm{\infty }}{\lim }\parallel S-\sum _{i=1}^n{v}_i\parallel =0;$$\lim_{n\to\infty} \left\| S - \sum_{i=1}^n v_i \right\| = 0 ;

we write

$$S=\sum _{i=1}^{\mathrm{\infty }}{v}_i$$S = \sum_{i=1}^\infty v_i

(with the right-hand side undefined if no such sum exists).

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

$$\parallel \sum _{i=1}^{\mathrm{\infty }}{v}_i\parallel \le \sum _{i=1}^{\mathrm{\infty }}\parallel v_i\parallel ,$$\left\| \sum_{i=1}^\infty v_i \right\| \leq \sum_{i=1}^\infty \|v_i\| ,

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).

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 distance $d(v,w)$ be

$$d(v,w)=\parallel w-v\parallel .$$d(v,w) = \|w - v\| .

Then $d$ is a pseudometric. Conversely, given any pseudometric $d$ on a vector space $V$, let $\parallel v\parallel$ be

$$\parallel v\parallel =d(0,v).$$\|v\| = d(0,v) .

Then $\parallel -\parallel$ satisfies the axioms (1–3) for a pseudonorm, except that it may satisfy (2) only for $r=\pm 1$. It will actually be a pseudonorm iff the pseudometric satisfies a homogeneity rule:

$$d(rv,rw)=\mid r\mid d(v,w).$$d(r v,r w) = |r| d(v,w) .

Thus pseudonorms correspond precisely to homogeneous pseudometrics.

Similarly, norms correspond to homogenous metrics and complete norms correspond to complete homogeneous metrics. Indeed, (1) says that the sequence of partial sums is a Cauchy sequence, while (2) 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 metric. Actually, one usually sees a sort of hybrid approach: a Banach space is a normed vector space whose corresponding metric is complete.

Morphisms of Banach spaces

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

• $\parallel f\parallel =\sup \{\parallel fv\parallel \mid \parallel v\parallel \le 1\}$;
• $\parallel f\parallel =\inf \{r\mid \forall v,\parallel fv\parallel \le r\parallel v\parallel \}$.

(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:

• $f$ is continuous (as measured by the pseudometrics on $V$ and $W$) at $0$;
• $f$ is continuous;
• $f$ is uniformly continuous;
• $\parallel f\parallel$ is finite.

In this case, we say that $f$ is bounded. (In constructive mathematics, it is necessary to further require that $\parallel f\parallel$ be a located real number.)

The bounded linear maps from $V$ to $W$ themselves form a pseudonormed vector space $ℬ(V,W)$. 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 $\mathrm{Ban}$ of Banach spaces is a closed category with $ℝ$ as the unit.

The clever reader will note that we have not yet defined $\mathrm{Ban}$ as a category! Naïvely, one might accept all bounded linear maps between Banach spaces as morphisms, but in the usual context this doesn't give the usual notion of isomorphism. Instead, we take a morphism to be a short linear map: a linear map $f$ such that $\parallel f\parallel \le 1$. Then the isomorphisms are the (surjective) linear isometries.

Note that this makes the ‘underlying set’ (in the sense of $\mathrm{Ban}$ as a concrete category like any closed category) of a Banach space its (closed) unit ball

$${\mathrm{Hom}}_{\mathrm{Ban}}(ℝ,V)\cong \{v\mid \parallel v\parallel \le 1\}$$Hom_Ban(\mathbb{R},V) \cong \{ v \;|\; \|v\| \leq 1 \}

rather than the set of all vectors in $V$ (the underlying set of $V$ as a vector space).

Examples

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

• ${ℝ}^n$ is a Banach space with

  $$\parallel (x_1,\dots ,x_n){\parallel }_p=\sqrt[p]{\sum _i\mid x_i{\mid }^p}.$$\|(x_1,\ldots,x_n)\|_p = \root p {\sum_i |x_i|^p} .

  (We can allow $p=\mathrm{\infty }$ by taking a limit; the result is that $\parallel x{\parallel }_{\mathrm{\infty }}={\max }_i\mid x_i\mid$.) 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.

• Let $l^p$ be the set of infinite sequences $(x_1,x_2,\dots )$ of real numbers such that

  $$\parallel (x_1,x_2,\dots ){\parallel }_p=\sqrt[p]{\sum _i\mid x_i{\mid }^p}$$\|(x_1,x_2,\ldots)\|_p = \root p {\sum_i |x_i|^p}

  exists as a finite real number. (The only question is whether the sum converges. Again $p=\mathrm{\infty }$ is a limit, with the result that $\parallel x{\parallel }_{\mathrm{\infty }}={\sup }_i\mid x_i\mid$.) Then $l^p$ is a Banach space with that norm. These are all versions of ${ℝ}^{\mathrm{\infty }}$, but they are no longer isomorphic for different values of $p$.

• More generally, let $A$ be any set and let $l^p(A)$ be the set of functions $f$ from $A$ to $ℝ$ such that

  $$\parallel f{\parallel }_p=\sqrt[p]{\sum _{x:A}\mid f(x){\mid }^p}$$\|f\|_p = \root p {\sum_{x: A} |f(x)|^p}

  exists as a finite real number. (Again, $\parallel f{\parallel }_{\mathrm{\infty }}={\sup }_{x:A}\mid f(x)\mid$.) Then $l^p(A)$ is a Banach space. (This example includes the previous examples, for $A$ a countable set.)

• On any measure space $X$, let ${ℒ}^p(X)$ be the set of measurable almost-everywhere-defined real-valued functions on $X$ such that

  $$\parallel f{\parallel }_p=\sqrt[p]{\int \mid f{\mid }^p}$$\|f\|_p = \root p {\int |f|^p}

  exists as a finite real number. (Again, the only question is whether the integral converges. And again $p=\mathrm{\infty }$ is a limit, with the result that $\parallel f{\parallel }_{\mathrm{\infty }}$ is the essential supremum of $\mid f\mid$.) As such, ${ℒ}^p(X)$ 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.)

• Any Hilbert space is Banach space; this includes all of the above examples for $p=2$.

Categorial operations on Banach spaces

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

• The category of Banach spaces admits small products. Given a small family of Banach spaces $\{X_{\alpha }{\}}_{\alpha \in A}$, its product in $\mathrm{Ban}$ is the subspace of the vector-space product

  $$\prod _{\alpha \in A}{X}_{\alpha }$$\prod_{\alpha \in A} X_\alpha

  consisting of $A$-tuples $⟨x_{\alpha }⟩$ which are uniformly bounded (i.e., there exists $C$ such that $\forall \alpha \in A:\parallel x_{\alpha }\parallel \le C$), taking the least such upper bound as the norm of $⟨x_{\alpha }⟩$. This norm is called the $\mathrm{\infty }$-norm; in particular, the product of an $A$-indexed family of copies of $ℝ$ or $ℂ$ is what is normally denoted as $l^{\mathrm{\infty }}(A)$.

• The category of Banach spaces admits equalizers. Indeed, the equalizer of a pair of maps $f,g:X\rightrightarrows Y$ in $\mathrm{Ban}$ 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).

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

  $$\underset{\alpha \in A}{⨁}{X}_{\alpha }$$\bigoplus_{\alpha \in A} X_\alpha

  with respect to the norm given by

  $$\parallel \underset{s\in S}{⨁}{x}_s\parallel =\sum _{s\in S}\parallel x_s\parallel ,$$\left\| \bigoplus_{s \in S} x_s \right\| = \sum_{s \in S} \|x_s\| ,

  where $S\subseteq A$ is finite and $\parallel x_s\parallel$ 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 $ℝ$ or $ℂ$ is what is normally denoted as $l^1(A)$.

• The category of Banach spaces admits coequalizers. 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.

To describe the tensor product of two Banach spaces (making $\mathrm{Ban}$ symmetric monoidal closed with respect to its usual internal hom), we invoke a standard consequence of the uniform boundedness principle?:

• Let $X,Y$ be Banach spaces, let $Z$ be a (pseudo)normed vector space, and suppose $f:X\times Y\to Z$ is separately linear and continuous (meaning each $f(x,-):Y\to Z$ and $f(-,y):X\to Z$ is linear and continuous). Then there is a uniform bound $C$ such that

  $$f(x,y)\le C\parallel x\parallel \cdot \parallel y\parallel .$$f(x, y) \leq C\|x\|\cdot\|y\| .

As a result, we may define $X{\otimes }_{\mathrm{Ban}}Y$ by completing the ordinary vector space tensor product with respect to a suitable norm. In detail, 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

$$\parallel \sum _{1\le i\le n}{a}_i(x_i\otimes y_i)\parallel =\sum _{1\le i\le n}\mid a_i\mid \parallel x_i\parallel \cdot \parallel y_i\parallel ,$$\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\| ,

and 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 $X{\otimes }_{\mathrm{Ban}}Y$.

To be described:

• duals ($p+q=pq$);
• completion ($\mathrm{Ban}$ is a reflective subcategory of $\mathrm{PsNVect}$).
• $\mathrm{Ban}$ as a (somewhat larger) category with duals.