A [vector space](/page/Vector%20Space) knows how to add and scale, but it does not yet know what it means for two vectors to be close.

That missing notion is fatal for analysis: without length there is no convergence, no continuity, no approximation, and no way to say that a sequence of functions is approaching a solution of an equation.

A normed space is the first place where linear algebra becomes geometric analysis. Finite-dimensional Euclidean space is the model example, but many of the most important spaces are infinite-dimensional: spaces of continuous functions, integrable functions, and bounded sequences. The same formal definition covers all of them, and that common language is powerful because many arguments depend only on three length axioms. At the same time, infinite-dimensional normed spaces behave in ways that finite-dimensional spaces hide, so this chapter is also the point where familiar geometry begins to acquire genuinely analytical depth.

## Definition

A usable notion of length on a vector space has to solve three problems at once: only the zero vector should have length zero, rescaling a vector should rescale its length by the absolute value of the scalar, and the direct route should not be longer than a route through an intermediate vector. These requirements are exactly the axioms that make linear estimates compatible with metric and topological arguments.

[definition: Normed Space] Let $V$ be a vector space over $\mathbb{R}$ or over $\mathbb{C}$. A norm on $V$ is a function $\|\cdot\|:V\to [0,\infty)$ such that, for all $x,y\in V$ and all scalars $\lambda$, the following conditions hold. The positivity condition is: if $\|x\|=0$, then $x=0$. The homogeneity identity is \begin{align*} \|\lambda x\|=|\lambda|\,\|x\|. \end{align*} The triangle inequality is \begin{align*} \|x+y\|&\leq \|x\|+\|y\|. \end{align*} A normed space is a pair $(V,\|\cdot\|)$ consisting of a vector space $V$ and a norm $\|\cdot\|$ on $V$. [/definition]

The real line gives the smallest nonzero example and fixes the connection with ordinary distance. It also helps separate the abstract notation from the familiar absolute value.

[example: The Absolute Value Norm] For $x\in\mathbb{R}$, define $\|x\|=|x|$. If $\|x\|=0$, then $|x|=0$, so $x=0$. For $\lambda,x\in\mathbb{R}$, the multiplicative property of absolute value gives \begin{align*} \|\lambda x\|=|\lambda x|=|\lambda|\,|x|=|\lambda|\,\|x\|. \end{align*} For $x,y\in\mathbb{R}$, the ordinary triangle inequality for absolute value gives \begin{align*} \|x+y\|=|x+y|\leq |x|+|y|=\|x\|+\|y\|. \end{align*} Thus $|\cdot|$ satisfies the norm axioms on $\mathbb{R}$. The metric induced by this norm is obtained by applying the norm to the difference: \begin{align*} d(x,y)=\|x-y\|=|x-y|. \end{align*} So the norm topology on $\mathbb{R}$ is the familiar topology coming from ordinary distance on the real line. [/example]

The first axiom says that the norm detects the zero vector. The second says that scaling a vector scales its length by the absolute value of the scalar. The third is the triangle inequality: it expresses the geometric idea that travelling from $0$ to $x+y$ through $x$ cannot be shorter than the direct route.

The notation $\|x\|$ depends on the chosen norm. The same vector space may carry many different norms, so when the norm matters it is written explicitly as $(V,\|\cdot\|)$.

## Metric Structure

### The Induced Distance

The norm immediately produces a distance function. This step matters because it lets every normed space inherit the language of metric spaces while retaining its linear operations.

[definition: Metric Induced by a Norm] Let $(V,\|\cdot\|)$ be a normed space. The metric induced by the norm is the function $d:V\times V\to [0,\infty)$ defined by \begin{align*} d:V\times V\to [0,\infty),\qquad (x,y)\mapsto \|x-y\|. \end{align*} [/definition]

The definition has now produced a candidate distance, but it still has to be checked before any metric language is legitimate. The next theorem is the bridge from linear length to topology: it says that the expression $\|x-y\|$ really does support open balls, convergence, and continuity. Positivity comes from positive definiteness, symmetry comes from homogeneity with scalar $-1$, and the triangle inequality for $d$ is the triangle inequality for the norm applied to $(x-z)+(z-y)$.

[quotetheorem:9982]

This result is the point where all metric-space language becomes available: one may now speak about open balls, convergent sequences, Cauchy sequences, continuity, and completeness in a normed space without adding any extra structure. The construction is also limited in an important way: it uses the norm, so changing the norm may change the metric unless the norms are known to be equivalent.

The induced metric still remembers the linear structure, and the next elementary result records exactly how the distance behaves under the two linear operations. This compatibility is what allows metric estimates to be translated and scaled without changing their form.

[quotetheorem:9983]

Thus a normed space is a [metric space](/page/Metric%20Space) with extra linear structure: translating the whole picture preserves distance, and scaling the picture multiplies every distance by the absolute value of the scalar.