Banach space

A Banach space is a normed vector space that is complete: every Cauchy sequence converges, in the norm, to an element of the space. Banach spaces are central objects of functional analysis, providing a setting for studying infinite-dimensional spaces of functions and sequences. Completeness makes it possible to construct elements through convergent approximations without leaving the chosen space. (ocw.mit.edu)

Definition and completeness

Let $X$ be a vector space over the real numbers or complex numbers. A norm is a function $x\mapsto\|x\|$ taking nonnegative real values and satisfying

The last condition is the triangle inequality. The norm defines a metric by

The pair $(X,\|\cdot\|)$ is a Banach space precisely when this metric makes $X$ a complete metric space. Explicitly, if for every $\varepsilon>0$ there is an $N$ such that $\|x_n-x_m\|<\varepsilon$ whenever $n,m\geq N$, then some $x\in X$ satisfies $\|x_n-x\|\to0$. (ocw.mit.edu)

An equivalent criterion is that every absolutely convergent vector series converges: whenever

the partial sums of $\sum_nu_n$ converge in $X$. This criterion translates a question about vector convergence into one about a numerical series. (bpb-us-w2.wpmucdn.com)

Standard examples

Every normed space of finite dimension over the real or complex numbers is complete. Thus $\mathbb R^n$ and $\mathbb C^n$ are Banach spaces with any norm, not only the Euclidean norm. (ocw.mit.edu)

Important infinite-dimensional examples include:

• Continuous-function spaces. The space $C([a,b])$ of continuous functions on a compact interval is complete under

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

  Convergence in this norm is uniform convergence.

• Sequence spaces. For $1\leq p<\infty$, the space $\ell^p$ consists of sequences satisfying $\sum_n|x_n|^p<\infty$, with norm $(\sum_n|x_n|^p)^{1/p}$. The space $\ell^\infty$ of bounded sequences is complete under the supremum norm. (ocw.mit.edu)

• Integrable-function spaces. The $L^p$ spaces, for $1\leq p\leq\infty$, are Banach spaces. Their elements identify measurable functions that agree almost everywhere. For finite $p$, the norm is defined using the Lebesgue integral:

  $$\|f\|_p=\left(\int|f|^p\,d\mu\right)^{1/p}.$$

  For $p=\infty$, the norm is the essential supremum. (bpb-us-w2.wpmucdn.com)

Every Hilbert space is a Banach space whose norm comes from an inner product. The converse is false: general Banach norms need not have this additional geometric structure. (bpb-us-w2.wpmucdn.com)

Dependence on the norm and basic constructions

Completeness concerns the chosen norm, not merely the underlying vector space. For example, $C([0,1])$ is complete in the supremum norm but incomplete in the norm $\int_0^1|f(t)|\,dt$. Continuous approximations to a step function can be Cauchy in the latter norm without converging to any continuous function. Equivalent norms, however, preserve completeness. (ocw.mit.edu)

A linear subspace of a Banach space is complete in the inherited norm exactly when it is closed. If $M$ is closed, the quotient space $X/M$ is Banach with

Without closedness, this expression may vanish on a nonzero coset. (ocw.mit.edu)

Every normed space has a completion that carries compatible vector operations and an extended norm, producing a Banach space. Nevertheless, completeness does not imply compactness: the closed unit ball of a normed space is compact in the norm topology exactly when the space is finite-dimensional. (ocw.mit.edu)

Operators and duality

A linear map $T:X\to Y$ between normed spaces is continuous exactly when it is bounded, meaning

for some constant $C$. Its operator norm is

The space of bounded linear maps $X\to Y$ is Banach whenever $Y$ is Banach, even if $X$ is incomplete. (live.ocw.mit.edu)

The continuous dual space $X^*$ consists of bounded scalar-valued linear functionals, with the operator norm. It is always Banach. The canonical map into the bidual,

is an isometric embedding. A Banach space is called reflexive when this map is onto. Reflexivity is stronger than completeness; for example, the Banach space of sequences tending to zero, with the supremum norm, is not reflexive. (live.ocw.mit.edu)

Fundamental theorems

Several major results organize Banach-space theory:

• The Hahn–Banach theorem extends a bounded linear functional from a subspace to the whole normed space without increasing its norm. Completeness is not required.
• The uniform boundedness principle states that a pointwise bounded family of bounded linear operators with Banach domain has uniformly bounded operator norms. (ocw.mit.edu)
• The open mapping theorem states that a bounded, surjective linear operator between Banach spaces is open. Consequently, a bounded bijective linear operator has a bounded inverse.
• The closed graph theorem states that an everywhere-defined linear operator between Banach spaces is bounded if its graph is closed in the product space. (ocw.mit.edu)

Completeness also supports the fixed-point theorem for a contraction mapping: successive iteration converges to a unique fixed point on a complete metric space. Applied to suitable function spaces, this gives existence and uniqueness results for differential equations. (ocw.mit.edu)