In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" in the real (physical) world. A norm is a real-valued function defined on the vector space that is commonly denoted ${\displaystyle x\mapsto \|x\|,}$ and has the following properties:

1. It is nonnegative, meaning that ${\displaystyle \|x\|\geq 0}$ for every vector ${\displaystyle x.}$
2. It is positive on nonzero vectors, that is,
3. For every vector ${\displaystyle x,}$ and every scalar ${\displaystyle \alpha ,}$
4. The triangle inequality holds; that is, for every vectors ${\displaystyle x}$ and ${\displaystyle y,}$

A norm induces a distance, called its (norm) induced metric, by the formula

which makes any normed vector space into a metric space and a topological vector space. If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true. For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm, but it is not complete for this norm.

An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of a Euclidean vector space is a special case that allows defining Euclidean distance by the formula

The study of normed spaces and Banach spaces is a fundamental part of functional analysis, which is a major subfield of mathematics.