Normed and Banach Spaces - Core Definitions

The concept of a normed space extends the familiar notion of distance and magnitude from Euclidean spaces to general vector spaces, providing the foundation for functional analysis.

DefinitionNormed Space

Let $X$ be a vector space over $\mathbb{K}$ (where $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$ ). A norm on $X$ is a function $\|\cdot\| : X \to [0,\infty)$ satisfying:

Positivity: $\|x\| = 0$ if and only if $x = 0$
Homogeneity: $\|\alpha x\| = |\alpha| \|x\|$ for all $\alpha \in \mathbb{K}$ , $x \in X$
Triangle Inequality: $\|x + y\| \leq \|x\| + \|y\|$ for all $x, y \in X$

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

Every norm induces a metric via $d(x,y) = \|x - y\|$ , making every normed space automatically a metric space. This connection allows us to discuss convergence, continuity, and completeness in normed spaces using the familiar language of metric spaces.

DefinitionBanach Space

A normed space $(X, \|\cdot\|)$ is called a Banach space if it is complete with respect to the metric induced by the norm. That is, every Cauchy sequence in $X$ converges to an element in $X$ .

ExampleStandard Examples of Normed Spaces

Euclidean Space: $\mathbb{R}^n$ with $\|x\|_2 = \sqrt{\sum_{i=1}^n |x_i|^2}$ is a Banach space
Sequence Spaces: For $1 \leq p < \infty$ , $\ell^p = \left\{(x_n) : \sum_{n=1}^\infty |x_n|^p < \infty\right\}$ with norm $\|(x_n)\|_p = \left(\sum_{n=1}^\infty |x_n|^p\right)^{1/p}$ is a Banach space
Function Spaces: $C[0,1]$ with supremum norm $\|f\|_\infty = \sup_{x \in [0,1]} |f(x)|$ is a Banach space
$L^p$ Spaces: For a measure space $(X, \mathcal{M}, \mu)$ and $1 \leq p < \infty$ , $L^p(X,\mu) = \left\{f : \int_X |f|^p \, d\mu < \infty\right\}$ with norm $\|f\|_p = \left(\int_X |f|^p \, d\mu\right)^{1/p}$ forms a Banach space

Remark

Not all norms are equivalent. On $\mathbb{R}^n$ , however, all norms are equivalent in the sense that they induce the same topology. This finite-dimensional phenomenon does not extend to infinite dimensions, where the choice of norm becomes crucial.

The structure of normed spaces allows us to develop a rich theory of linear operators, dual spaces, and spectral analysis, forming the backbone of modern functional analysis and its applications to differential equations, quantum mechanics, and optimization theory.