Normed & Banach Spaces

A normed space keeps the algebra of a vector space and adds a quantitative notion of size. This makes approximation, continuity, error, and infinite sums available in the same language. Completeness is the structural threshold: many fundamental operator theorems work only when limits of admissible approximations remain in the space.

Norms and their induced metrics #

A norm on a real or complex vector space $X$ satisfies

$$\|x\|=0\iff x=0,\qquad \|\alpha x\|=|\alpha|\|x\|, \qquad \|x+y\|\le\|x\|+\|y\|.$$

It induces the translation-invariant metric $d(x,y)=\|x-y\|$. A Banach space is a normed space complete in this metric. The scalar fields $\mathbb R$ and $\mathbb C$, every finite-dimensional normed space, $C(K)$ with the sup norm, and $\ell^p,L^p$ for $1\le p\le\infty$ are Banach spaces.

The same vector space can behave differently under different norms. On $C^1[0,1]$, the sup norm $\|f\|_\infty$ is incomplete because a uniform limit of differentiable functions need not be differentiable. The norm

$$\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$$

controls both function and derivative and makes $C^1[0,1]$ complete.

Equivalent norms #

Norms $\|\cdot\|_a$ and $\|\cdot\|_b$ are equivalent if constants $c,C>0$ satisfy

$$c\|x\|_a\le\|x\|_b\le C\|x\|_a\quad\text{for every }x.$$

Equivalent norms have the same convergent and Cauchy sequences, topology, closed sets, and completeness. All norms on a finite-dimensional space are equivalent. In infinite dimensions this fails: the sup and $L^1$ norms on continuous functions are not equivalent because narrow spikes can have fixed height and vanishing area.

Series criterion for completeness #

A normed space $X$ is Banach iff every series satisfying

$$\sum_{n=1}^{\infty}\|x_n\|<\infty$$

converges in $X$. Absolute summability makes partial sums Cauchy by the triangle inequality. The converse can be proved by extracting a rapidly Cauchy subsequence and expressing its increments as an absolutely summable series. This criterion is the engine behind Neumann-series inversion and many operator constructions.

Subspaces, quotients, and products #

A linear subspace of a Banach space is Banach under the inherited norm iff it is closed. Finite-dimensional subspaces are always closed. The quotient $X/M$ by a closed subspace $M$ becomes Banach with

$$\|x+M\|=\inf_{m\in M}\|x-m\|.$$

The infimum measures distance to the entire equivalence class. Closedness of $M$ is what prevents a nonzero coset from having zero quotient norm.

Finite products of Banach spaces are Banach under, for example, $\|(x,y)\|=\|x\|+\|y\|$ or $\max(\|x\|,\|y\|)$; these product norms are equivalent.

Dense cores and completions #

A dense subspace may contain the objects convenient for computation while its completion contains the limits needed for analysis. Finite-support sequences $c_{00}$ are dense in $\ell^p$ for $p<\infty$ but not in $\ell^\infty$. Polynomials are dense in $C[a,b]$ under the sup norm. Smooth compactly supported functions are dense in many $L^p$ and Sobolev spaces under the appropriate hypotheses.

Why completeness appears in operator theorems #

Baire category holds in complete metric spaces: a nonempty complete space cannot be a countable union of closed sets with empty interior. Uniform boundedness and open mapping turn this qualitative fact into quantitative operator bounds. If completeness is silently dropped, pointwise-bounded families can fail to be uniformly bounded and bijective bounded maps can have unbounded inverses.

When a proof invokes an infinite sum, an inverse constructed by iteration, or a Baire-category argument, check completeness at that exact step rather than treating “Banach” as decorative terminology.