Analysis requires a way to measure the "size" of the objects it studies — [functions](/page/Function), [sequences](/page/Sequence), operators, distributions. A **norm** is the abstraction of size: it assigns to each element of a vector space a non-negative real number satisfying the triangle inequality, positive-definiteness, and homogeneity. Equipped with a norm, a vector space becomes a **normed vector space** — a setting where notions of convergence, [continuity](/page/Continuity), and completeness are governed by a single, explicit quantity $\|x\|$.

Normed spaces sit at a critical junction in the hierarchy of mathematical structures. They are special cases of [metric spaces](/page/Metric%20Space) (the metric is $d(x,y) = \|x - y\|$), of [topological vector spaces](/page/Topological%20Vector%20Space) (the topology is the metric topology), and of locally convex spaces (a single norm is a single seminorm). But they carry more structure than any of these: the norm is compatible with the linear structure in a way that a general metric or topology need not be, and this compatibility is what makes the theories of [bounded linear operators](/page/Linear%20Operators%20on%20Banach%20Spaces), [dual spaces](/page/Topological%20Dual), and spectral theory possible.

The central dichotomy in the theory is between **complete** normed spaces ([Banach spaces](/page/Banach%20Space)) and incomplete ones. Completeness — the convergence of every Cauchy sequence — is the property that makes [limits](/page/Limit) exist, and its presence or absence controls whether fundamental theorems like the [Open Mapping Theorem](/theorems/631), the [Closed Graph Theorem](/theorems/217), and the [Uniform Boundedness Principle](/theorems/549) hold.

[motivation] ## Motivation ### From Absolute Value to Norm On $\mathbb{R}$, the absolute value $|x|$ measures the size of a number and gives a distance $d(x, y) = |x - y|$. On $\mathbb{R}^n$, the Euclidean length $|x| = (\sum_{i=1}^n x_i^2)^{1/2}$ does the same. Both satisfy the triangle inequality, scale linearly ($|\alpha x| = |\alpha| |x|$), and vanish only at the origin. The question is: can we extend this structure to infinite-dimensional vector spaces — spaces of functions, sequences, or distributions — in a way that supports the convergence arguments of analysis? A **metric** alone is not enough. Metrics on vector spaces need not be compatible with the linear structure: there exist translation-invariant metrics on $\mathbb{R}^\infty$ for which scalar multiplication is discontinuous. A **topology** alone is too general: the topology on $\mathcal{D}(\Omega)$ ([test functions](/page/Test%20Function)) cannot be generated by any single norm, and the full machinery of [topological vector spaces](/page/Topological%20Vector%20Space) is needed to handle such spaces. What we want is the simplest structure that simultaneously provides a notion of size, induces a topology compatible with the algebra, and supports quantitative estimates — bounds of the form $\|Tx\| \le C\|x\|$ that are the language of analysis. ### What a Norm Must Do A norm on a vector space $X$ is a function $\|\cdot\|: X \to [0, \infty)$ that captures the essential properties of length: it is zero only at the origin (so distinct points are distinguishable), it scales linearly with the scalar field (so $\|2x\| = 2\|x\|$), and it satisfies the triangle inequality (so the "direct path" is always shortest). These three axioms are precisely what is needed to define a metric $d(x,y) = \|x - y\|$ that is translation-invariant ($d(x+z, y+z) = d(x,y)$) and homogeneous ($d(\alpha x, \alpha y) = |\alpha| d(x,y)$) — and these two properties ensure that the induced topology makes addition and scalar multiplication continuous, so the resulting topological vector space is well-behaved. ### The Completeness Question Having a norm gives us [Cauchy sequences](/page/Cauchy%20Sequence). But Cauchy sequences need not converge: the rational numbers $\mathbb{Q}$ with the absolute value form a normed space in which $\sum_{k=0}^n 1/k!$ is Cauchy but does not converge (its limit is $e \notin \mathbb{Q}$). In infinite-dimensional function spaces, the same phenomenon is more severe: a sequence of smooth functions can be Cauchy in one norm but converge to a non-smooth limit (or to no element of the space at all). The question of whether every Cauchy sequence converges — **completeness** — is the dividing line between spaces where analysis works well (Banach spaces) and spaces where it does not. [/motivation]

## Definition

A norm must capture three properties of the absolute value: positive-definiteness (only zero has zero size), absolute homogeneity (scaling the vector scales the size), and the triangle inequality (the size of a sum is at most the sum of the sizes). These are the minimal axioms that produce a useful metric from a linear structure.

[definition:Norm] Let $X$ be a vector space over $\mathbb{F}$, where $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$. A **norm** on $X$ is a function $\|\cdot\|: X \to [0, \infty)$ satisfying: 1. **Positive-definiteness:** $\|x\| = 0$ if and only if $x = 0$. 2. **Absolute homogeneity:** $\|\alpha x\| = |\alpha| \|x\|$ for all $\alpha \in \mathbb{F}$ and $x \in X$. 3. **Triangle inequality:** $\|x + y\| \le \|x\| + \|y\|$ for all $x, y \in X$. [/definition]

[definition:Normed Vector Space] A **normed vector space** (or **normed space**) is a pair $(X, \|\cdot\|)$ consisting of a vector space $X$ over $\mathbb{F}$ and a norm $\|\cdot\|$ on $X$. [/definition]

Every norm induces a metric $d: X \times X \to [0, \infty)$ by $d(x, y) := \|x - y\|$. This metric is translation-invariant ($d(x+z, y+z) = d(x,y)$) and absolutely homogeneous ($d(\alpha x, \alpha y) = |\alpha| d(x,y)$). Conversely, a translation-invariant, absolutely homogeneous metric on a vector space arises from a norm via $\|x\| = d(x, 0)$.

The metric induces a topology (the metric topology), making $(X, \|\cdot\|)$ a [topological vector space](/page/Topological%20Vector%20Space). This topology is Hausdorff (distinct points can be separated by open sets, since $\|x - y\| > 0$ for $x \neq y$), first countable (open balls $B(x, 1/n)$ form a countable neighbourhood basis), and locally convex (the open balls are convex [sets](/page/Set)). The open ball of radius $r$ centred at $x$ is $B(x, r) := \{y \in X : \|y - x\| < r\}$.

## Examples

[example:The Euclidean Norm On $\mathbb{R}^n$] The **Euclidean norm** on $\mathbb{R}^n$ is \begin{align*} \|x\|_2 := \left(\sum_{i=1}^n x_i^2\right)^{1/2}. \end{align*} Positive-definiteness and homogeneity are immediate. The triangle inequality $\|x + y\|_2 \le \|x\|_2 + \|y\|_2$ is the **Cauchy-Schwarz inequality** applied to the inner product $x \cdot y = \sum_{i=1}^n x_i y_i$: since $(x \cdot y)^2 \le \|x\|_2^2 \|y\|_2^2$, one obtains $\|x + y\|_2^2 = \|x\|_2^2 + 2(x \cdot y) + \|y\|_2^2 \le (\|x\|_2 + \|y\|_2)^2$. More generally, $\mathbb{R}^n$ carries the family of **$p$-norms** for $1 \le p \le \infty$: \begin{align*} \|x\|_p &:= \left(\sum_{i=1}^n |x_i|^p\right)^{1/p} \quad (1 \le p < \infty), \\ \|x\|_\infty &:= \max_{1 \le i \le n} |x_i|. \end{align*} The triangle inequality for $\|\cdot\|_p$ is the **Minkowski inequality**, which follows from the **Hölder inequality** $\sum |x_i y_i| \le \|x\|_p \|y\|_q$ with $1/p + 1/q = 1$. [/example]

[example:Sequence Spaces $\ell^p$] For $1 \le p \le \infty$, the **sequence space** $\ell^p$ consists of sequences $x = (x_k)_{k=1}^\infty$ in $\mathbb{C}$ with finite $p$-norm: \begin{align*} \ell^p &:= \left\{(x_k) \in \mathbb{C}^{\mathbb{N}} : \|x\|_{\ell^p} := \left(\sum_{k=1}^\infty |x_k|^p\right)^{1/p} < \infty\right\} \quad (1 \le p < \infty), \\ \ell^\infty &:= \left\{(x_k) \in \mathbb{C}^{\mathbb{N}} : \|x\|_{\ell^\infty} := \sup_{k \ge 1} |x_k| < \infty\right\}. \end{align*} Each $\ell^p$ is a Banach space (completeness is proved below). The inclusions $\ell^p \subset \ell^q$ for $p \le q$ hold with $\|x\|_{\ell^q} \le \|x\|_{\ell^p}$. The space $\ell^2$ is a [Hilbert space](/page/Hilbert%20Space) with inner product $(x, y)_{\ell^2} = \sum_{k=1}^\infty x_k \overline{y_k}$. [/example]

[example:Function Spaces $L^p$] For a measure space $(E, \mathcal{E}, \mu)$ and $1 \le p \le \infty$, the **Lebesgue space** $L^p(E, \mu)$ consists of (equivalence classes of) measurable functions $f: E \to \mathbb{C}$ with finite $p$-norm: \begin{align*} \|f\|_{L^p} &:= \left(\int_E |f|^p \, d\mu\right)^{1/p} \quad (1 \le p < \infty), \\ \|f\|_{L^\infty} &:= \operatorname{ess\,sup}_{x \in E} |f(x)| = \inf\{C \ge 0 : |f(x)| \le C \text{ $\mu$-a.e.}\}. \end{align*} Two functions $f, g$ with $f = g$ $\mu$-a.e. are identified, so elements of $L^p$ are equivalence classes — this is necessary for positive-definiteness ($\|f\|_{L^p} = 0$ implies $f = 0$ a.e., hence $f = 0$ in $L^p$). The triangle inequality for $L^p$ is the **Minkowski inequality** for integrals. Each $L^p$ space is complete — the Riesz-Fischer theorem — making them Banach spaces. The space $L^2(E, \mu)$ is a Hilbert space with inner product $(f, g)_{L^2} = \int_E f \bar{g} \, d\mu$. For $1 \le p < \infty$ and $E \subseteq \mathbb{R}^n$ open, $L^p(E, \mathcal{L}^n)$ is [separable](/theorems/548). The space $L^\infty$ is not [separable](/page/Separable) when $\mu$ is non-atomic. [/example]

[example:The Supremum Norm On $C(K)$] For a compact Hausdorff space $K$, the space $C(K)$ of continuous functions $f: K \to \mathbb{C}$ carries the **supremum norm** \begin{align*} \|f\|_\infty := \max_{x \in K} |f(x)|. \end{align*} The maximum exists by compactness and continuity. This makes $C(K)$ a Banach space: if $(f_n)$ is Cauchy in $C(K)$, the pointwise limit $f(x) = \lim_n f_n(x)$ exists (since $\mathbb{C}$ is complete) and is continuous (since the convergence is uniform), so $f \in C(K)$ and $\|f_n - f\|_\infty \to 0$. For $K = [0,1]$, the [topological dual](/page/Topological%20Dual) $C([0,1])^*$ is the space of complex Radon measures on $[0,1]$. [/example]

[example:Sobolev Norms] The [Sobolev space](/page/Sobolev%20Space) $W^{k,p}(U)$ for an open set $U \subseteq \mathbb{R}^n$ carries the norm \begin{align*} \|u\|_{W^{k,p}(U)} := \left(\sum_{|\alpha| \le k} \|D^\alpha u\|_{L^p(U)}^p\right)^{1/p} \quad (1 \le p < \infty), \end{align*} which measures both the function and its weak [derivatives](/page/Derivative) up to order $k$ in the $L^p$ sense. The Sobolev norm is strictly stronger than the $L^p$ norm: $\|u\|_{L^p} \le \|u\|_{W^{k,p}}$. The space $W^{k,p}(U)$ is a Banach space, and $H^k(U) = W^{k,2}(U)$ is a Hilbert space with inner product $(u, v)_{H^k} = \sum_{|\alpha| \le k} (D^\alpha u, D^\alpha v)_{L^2}$. [/example]

## Completeness and Banach Spaces

In a normed space, every convergent sequence is Cauchy (since $\|x_m - x_n\| \le \|x_m - x\| + \|x - x_n\| \to 0$). The converse — every Cauchy sequence converges — need not hold, and its validity is the most important structural property a normed space can have.

[definition:Banach Space] A normed vector space $(X, \|\cdot\|)$ is a **Banach space** if it is **complete**: every Cauchy sequence in $X$ converges to an element of $X$. Explicitly, if $(x_n)_{n=1}^\infty \subset X$ satisfies $\|x_m - x_n\| \to 0$ as $m, n \to \infty$, then there exists $x \in X$ with $\|x_n - x\| \to 0$. [/definition]

Completeness is not an intrinsic property of the vector space — it depends on the norm. The same vector space can be complete in one norm and incomplete in another.