Normed and Banach Spaces - Key Properties

Understanding convergence in normed spaces is fundamental to functional analysis. The concept of completeness distinguishes Banach spaces as the most well-behaved normed spaces.

DefinitionConvergence in Normed Spaces

Let $(X, \|\cdot\|)$ be a normed space. A sequence $(x_n)$ in $X$ converges to $x \in X$ if $\lim_{n \to \infty} \|x_n - x\| = 0$ We write $x_n \to x$ or $\lim_{n \to \infty} x_n = x$ .

A sequence $(x_n)$ is Cauchy if for every $\varepsilon > 0$ , there exists $N \in \mathbb{N}$ such that $\|x_m - x_n\| < \varepsilon \quad \text{for all } m, n \geq N$

In any normed space, every convergent sequence is Cauchy (this follows from the triangle inequality). However, the converse is not always true, and this leads to the crucial notion of completeness.

DefinitionComplete Normed Spaces

A normed space $(X, \|\cdot\|)$ is complete if every Cauchy sequence in $X$ converges to a limit in $X$ . A complete normed space is called a Banach space.

ExampleComplete and Incomplete Spaces

Complete: The space $\ell^2$ of square-summable sequences with norm $\|(x_n)\|_2 = \sqrt{\sum_{n=1}^\infty |x_n|^2}$ is a Banach space
Incomplete: The space $C[0,1]$ with the $L^1$ norm $\|f\|_1 = \int_0^1 |f(x)| \, dx$ is NOT complete. Cauchy sequences may converge to discontinuous functions
Completion: The space of polynomials on $[0,1]$ with supremum norm is not complete, but its completion is $C[0,1]$

Remark

Every normed space can be "completed" by adding limits of Cauchy sequences. This completion process is unique up to isometry and produces a Banach space containing the original space as a dense subspace.

The importance of completeness cannot be overstated. Many fundamental theorems in functional analysis—such as the Banach Fixed Point Theorem, the Open Mapping Theorem, and the Closed Graph Theorem—require completeness in their hypotheses.

ExampleSeries in Banach Spaces

In a Banach space $X$ , a series $\sum_{n=1}^\infty x_n$ converges if and only if the sequence of partial sums $s_n = \sum_{k=1}^n x_k$ converges in $X$ . The series converges absolutely if $\sum_{n=1}^\infty \|x_n\| < \infty$ .

A crucial fact: In a Banach space, absolute convergence implies convergence. This property characterizes Banach spaces among normed spaces.

The interplay between algebraic structure (vector space operations) and topological structure (norm-induced convergence) makes Banach spaces the natural setting for infinite-dimensional analysis.