Lecture 9: Banach Spaces

In an inner product space, the inner product determines the length of a vector. More generally, we can specify the length directly by defining a norm.

Definition (1.0) — Normed Space

Let $V$ be a vector space over $\mathbb{F}$, where $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$. A norm on $V$ is a function $\|\cdot\|\colon V\to[0,\infty)$ satisfying the following properties for all $\mathbf{u},\mathbf{v}\in V$ and $\alpha\in\mathbb{F}$.

Positive definiteness. $\|\mathbf{v}\|\ge 0$, with equality if and only if $\mathbf{v}=\mathbf{0}$.
Absolute homogeneity. $\|\alpha\mathbf{v}\|=|\alpha|\,\|\mathbf{v}\|$.
Triangle inequality. $\|\mathbf{u}+\mathbf{v}\|\le\|\mathbf{u}\|+\|\mathbf{v}\|$.

The pair $(V,\|\cdot\|)$ is called a normed space.

Every norm induces a metric on $V$ given by

$$ d(\mathbf{u},\mathbf{v})=\|\mathbf{u}-\mathbf{v}\|. $$

Definition (1.1) — Banach Space

A Banach space is a normed space $(V,\|\cdot\|)$ in which every Cauchy sequence converges to a vector in $V$. A sequence $(\mathbf{v}_n)$ is Cauchy if for every $\epsilon>0$, there exists $N\in\mathbb{N}$ such that $\|\mathbf{v}_n-\mathbf{v}_m\|\lt\epsilon$ for all $m,n\ge N$.

Example 1

Let $C[a,b]$ be the space of continuous functions $f\colon[a,b]\to\mathbb{F}$. Its supremum norm is

$$ \|f\|_\infty = \sup_{x\in[a,b]}|f(x)| = \max_{x\in[a,b]}|f(x)|. $$

Every Cauchy sequence in this norm converges uniformly to a function on $[a,b]$. The uniform limit of continuous functions is continuous, so the limit belongs to $C[a,b]$. Thus $C[a,b]$ is a Banach space.

Example 2

Let $(H,\langle\cdot,\cdot\rangle)$ be a Hilbert space. Its inner product induces the norm

$$ \|\mathbf{v}\|=\sqrt{\langle\mathbf{v},\mathbf{v}\rangle}. $$

A Hilbert space is complete in this norm by definition. Hence every Hilbert space is a Banach space.

Example 3

Let $\mathbb{T}=\mathbb{R}/(2\pi\mathbb{Z})$ denote the circle. The space $L^1(\mathbb{T})$ consists of equivalence classes of measurable functions $f\colon\mathbb{T}\to\mathbb{C}$ satisfying

$$ \int_{-\pi}^{\pi}|f(x)|\,dx \lt \infty. $$

Two functions represent the same element if they agree almost everywhere. With the normalized norm

$$ \|f\|_{L^1} = \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|\,dx, $$

$L^1(\mathbb{T})$ is a Banach space. Its norm fails the parallelogram identity, so no inner product induces this norm.

Example 4

The space $\ell^1(\mathbb{N})$ consists of all absolutely summable sequences. It is given by

$$ \ell^1(\mathbb{N}) = \left\{ (c_n)_{n=1}^{\infty}\in\mathbb{F}^{\mathbb{N}} \,\middle|\, \sum_{n=1}^{\infty}|c_n| \lt \infty \right\}. $$

It is a Banach space with the norm

$$ \|(c_n)\|_1=\sum_{n=1}^{\infty}|c_n|. $$

Example 5

The space $c_0$ consists of all sequences that converge to zero. It is given by

$$ c_0 = \left\{ (c_n)_{n=1}^{\infty}\in\mathbb{F}^{\mathbb{N}} \,\middle|\, \lim_{n\to\infty}c_n=0 \right\}. $$

Every sequence that converges to zero is bounded, so its supremum norm is finite. With the norm

$$ \|(c_n)\|_\infty=\sup_{n\in\mathbb{N}}|c_n|, $$

$c_0$ is a Banach space. Indeed, $c_0$ is a closed subspace of the Banach space of bounded sequences in the next example.

Example 6

The space of bounded sequences is

$$ \ell^\infty(\mathbb{N}) = \left\{ (c_n)_{n=1}^{\infty}\in\mathbb{F}^{\mathbb{N}} \,\middle|\, \sup_{n\in\mathbb{N}}|c_n| \lt \infty \right\}. $$

Its norm is

$$ \|(c_n)\|_\infty=\sup_{n\in\mathbb{N}}|c_n|. $$

To see why this space is complete, let $x^{(k)}=(x_n^{(k)})_{n=1}^{\infty}$ be Cauchy in the supremum norm. For each fixed $n$, the scalars $x_n^{(k)}$ are Cauchy and have a limit $x_n$. Given $\epsilon>0$, choose $K$ so that $\|x^{(k)}-x^{(j)}\|_\infty \lt \epsilon/2$ for all $k,j\ge K$. Letting $j\to\infty$ in each coordinate gives

$$ |x_n^{(k)}-x_n|\le\epsilon/2 \qquad \text{for every } n\in\mathbb{N} \text{ and } k\ge K. $$

Taking $k=K$ shows that $x=(x_n)$ is bounded, since $x^{(K)}$ is bounded. Taking the supremum over $n$ shows that $\|x^{(k)}-x\|_\infty\to 0$. Thus $\ell^\infty(\mathbb{N})$ is a Banach space.

References

SFSU Fourier Analysis by Professor Chun-Kit Lai
Fourier Analysis: An Introduction by Elias M.Stein and Rami Shakarchi