Banach Spaces — Mathematics Knowledge System

Overview

A Banach space is a normed vector space that is complete: every Cauchy sequence converges to a limit inside the space. Completeness is the key analytic property that enables fixed-point theorems, the open mapping theorem, and the uniform boundedness principle. The most important examples are the L^p spaces of integrable functions, the space C[0,1] of continuous functions with the sup-norm, and all finite-dimensional normed spaces.

Intuition

Completeness means there are no 'missing limits': if a sequence of vectors gets closer and closer together, it must converge to something in the space. Without completeness, calculus breaks down — limits of well-behaved sequences might fall outside the space. Think of the rational numbers: Cauchy sequences of rationals can converge to irrationals (which are 'missing'), making Q incomplete. The real numbers are the completion of Q, just as L^p spaces are completions of spaces of step functions.

Formal Definition

Definition

A sequence (x_n) in a normed space X is Cauchy if for every epsilon > 0 there exists N such that ||x_m - x_n|| < epsilon for all m,n > N. A normed space X is complete, or a Banach space, if every Cauchy sequence in X converges to a limit in X. The completion of any normed space X is a Banach space X-bar containing X as a dense subspace.

\forall \varepsilon>0\; \exists N:\; m,n>N \implies \|x_m - x_n\| < \varepsilon

Cauchy condition

\ell^p = \left\{ (x_n)_{n\ge 1} : \sum_{n=1}^\infty |x_n|^p < \infty \right\},\quad \|(x_n)\|_p = \left(\sum_{n=1}^\infty |x_n|^p\right)^{1/p}

l^p sequence space

X/Y = \{ x + Y : x \in X \},\quad \|x+Y\|_{X/Y} = \inf_{y \in Y}\|x-y\|

Quotient space norm

Notation

Notation	Meaning
$\mathcal{B}(X)$	Algebra of bounded linear operators on X
$X/Y$	Quotient of Banach space X by closed subspace Y
$\ell^p$	Space of p-summable sequences
$c_0$	Space of sequences converging to zero, with sup-norm

Properties

Closed subspaces inherit completeness

A closed subspace of a Banach space is a Banach space.

Finite products of Banach spaces are Banach

If X_1, ..., X_n are Banach spaces then X_1 x ... x X_n with any p-norm is a Banach space.

Theorems

Theorem 2.1: Theorem 2.1

The space C[0,1] of continuous real-valued functions on [0,1] equipped with the sup-norm ||f||_inf = max_{t in [0,1]} |f(t)| is a Banach space.

Theorem 2.2: Theorem 2.2

If X is a Banach space and Y is a closed subspace of X, then the quotient space X/Y equipped with the quotient norm ||x+Y|| = inf_{y in Y} ||x-y|| is a Banach space.

Theorem 2.3: Series Characterisation of Completeness

A normed space X is complete if and only if every absolutely convergent series converges: sum ||x_n|| < infinity implies sum x_n converges in X.

Worked Examples

1
Let (x^{(k)}) be a Cauchy sequence in c_0, where x^{(k)} = (x^{(k)}_n)_{n>=1}.
2
For each fixed n, the scalars x^{(k)}_n form a Cauchy sequence in R, hence converge: x^{(k)}_n -> x_n as k -> infinity.
3
Since the convergence is uniform (Cauchy in sup-norm), x^{(k)} -> x = (x_n) uniformly.
$\|x^{(k)} - x\|_\infty = \sup_n |x^{(k)}_n - x_n| \to 0$
4
It remains to show x in c_0. Given epsilon > 0, choose k so ||x^{(k)} - x||_inf < epsilon/2. Since x^{(k)} in c_0, choose N so |x^{(k)}_n| < epsilon/2 for n > N. Then |x_n| <= |x_n - x^{(k)}_n| + |x^{(k)}_n| < epsilon for n > N. Hence x_n -> 0.

✓ Answer

c_0 is complete: Cauchy sequences converge uniformly to a limit sequence that also tends to zero.

Practice Problems

Mediumproof writing

Use the series characterisation of completeness to prove that R^n (with any norm) is complete.

Mediumfree response

Let Y be a dense subspace of a Banach space X, and let T: Y -> Z be a bounded linear operator into a Banach space Z. Show that T extends uniquely to a bounded linear operator T-bar: X -> Z with the same norm.

MediumMultiple choice

Which of the following is a Banach space?

The space of polynomials on [0,1] with the sup-normThe space of rational-valued continuous functions on [0,1] with the sup-normC[0,1] with the sup-normThe space of smooth functions C^inf[0,1] with the sup-norm

Common Mistakes

Common Mistake

Thinking that every normed space is complete.

Completeness is an extra condition. The rationals Q, polynomials with the sup-norm, and many function spaces are incomplete normed spaces.

Common Mistake

Assuming a Cauchy sequence is convergent in any normed space.

Cauchy sequences converge only if the space is complete. In an incomplete space, Cauchy sequences may not converge.

Quiz

Historical Background

Stefan Banach introduced complete normed spaces in his 1920 Lwów doctoral thesis (published 1922). The name 'Banach space' was coined by Maurice Fréchet. Banach's 1932 book Théorie des opérations linéaires, written with contributions from the entire Lwów School of Mathematics (Mazur, Schauder, Ulam, and others), remains a landmark in twentieth-century mathematics. Hans Hahn and Norbert Wiener independently obtained related results around the same period.

1922
Banach's thesis published, defining complete normed spaces
Stefan Banach
1927
Hahn proves the Hahn-Banach theorem independently
Hans Hahn
1932
Théorie des opérations linéaires published
Stefan Banach
1936
Clarkson introduces uniformly convex Banach spaces and proves reflexivity of L^p
James Clarkson

Summary

A Banach space is a complete normed vector space: every Cauchy sequence converges within the space.
Key examples include C[0,1] with the sup-norm, l^p sequence spaces, and L^p function spaces.
The series characterisation: completeness is equivalent to every absolutely convergent series converging.
Quotient spaces X/Y are Banach when X is Banach and Y is a closed subspace.
Dense incomplete subspaces (like polynomials inside C[0,1]) illustrate that completeness is a delicate property.

References

BookRudin, W. — Functional Analysis (2nd ed.), Chapter 2
BookConway, J. — A Course in Functional Analysis, Chapter 1
Websiteen.wikipedia.org