# Normed vector space

A normed vector space is a vector space over the real or complex numbers equipped with a norm, a function that assigns to each vector a nonnegative length satisfying three axioms: only the zero vector has length zero, scaling a vector scales its length by the absolute value of the scalar, and the triangle inequality \( \lVert x+y\rVert \le \lVert x\rVert+\lVert y\rVert \) holds for all vectors.<sup>[1](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup> The norm generalizes the absolute value on \( \mathbb{R} \) and the modulus on \( \mathbb{C} \), and it makes the algebraic structure of a vector space compatible with analytic notions of distance, convergence, and completeness.<sup>[1](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf)</sup>

| Key fact | Detail |
|---|---|
| Norm axioms | \( \lVert x\rVert = 0 \) only for \( x=0 \); \( \lVert \lambda x\rVert = \lvert\lambda\rvert\,\lVert x\rVert \); \( \lVert x+y\rVert \le \lVert x\rVert+\lVert y\rVert \)<sup>[1](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf)</sup> |
| Induced metric | \( d(x,y)=\lVert x-y\rVert \) is a metric, so every normed space is a metric and topological vector space<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup> |
| Banach space | A normed space complete under this metric; every normed space has a Banach completion<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup> |
| Finite dimension | Every finite-dimensional normed space is complete, and all norms on it are equivalent<sup>[3](https://www.maths.ox.ac.uk/system/files/attachments/lecture-notes-MT-2018-Nov-24.pdf)</sup> |
| Inner product case | A norm arises from an inner product exactly when it satisfies the parallelogram law \( \lVert x+y\rVert^2+\lVert x-y\rVert^2 = 2(\lVert x\rVert^2+\lVert y\rVert^2) \)<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup> |
| History | Banach's 1920 dissertation, published in 1922, gave the axiomatic definition; the name "Banach space" was coined by Fréchet<sup>[4](https://mathshistory.st-andrews.ac.uk/Biographies/Banach/)</sup> |

## Definition

Let \( X \) be a vector space over a field \( \mathbb{K} \) equal to \( \mathbb{R} \) or \( \mathbb{C} \). A norm is a function \( \lVert\cdot\rVert : X \to [0,\infty) \) with three properties, called the axiom of identity, the axiom of homogeneity, and the triangle inequality:<sup>[5](https://courses.ms.ut.ee/2024/fa1/spring/uploads/Main/FA1--Lecture_Notes--2024-02-19.pdf)</sup>

1. \( \lVert x\rVert = 0 \) if and only if \( x = 0 \);
2. \( \lVert \alpha x\rVert = \lvert\alpha\rvert\,\lVert x\rVert \) for every scalar \( \alpha \);
3. \( \lVert x+y\rVert \le \lVert x\rVert+\lVert y\rVert \) for all \( x, y \).

A function satisfying the last two but possibly not the first is a seminorm; seminorms that do not separate points arise when functions are identified up to sets of measure zero, and the spaces \( L^1 \), \( L^2 \), and \( L^\infty \) are built from such pseudonorms by passing to equivalence classes.<sup>[1](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf)</sup><sup> • </sup><sup>[6](https://www.math.stonybrook.edu/~aknapp/download/b2-realanal-inside.pdf)</sup> A linear subspace of a normed space is again a normed space under the same norm.<sup>[5](https://courses.ms.ut.ee/2024/fa1/spring/uploads/Main/FA1--Lecture_Notes--2024-02-19.pdf)</sup>

## The induced metric and topology

Every norm defines a metric by \( d(x,y) = \lVert x-y\rVert \), and this metric is compatible with the linear structure: it is translation invariant, and dilations scale distances linearly.<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup><sup> • </sup><sup>[7](https://adwendland.github.io/applied-mathematics.pdf)</sup> The metric in turn defines a topology, so a normed space carries the natural structure of a topological vector space in which addition, scalar multiplication, and the norm itself are continuous.<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup><sup> • </sup><sup>[5](https://courses.ms.ut.ee/2024/fa1/spring/uploads/Main/FA1--Lecture_Notes--2024-02-19.pdf)</sup> Convergence \( x_n \to x \) means \( \lVert x_n - x\rVert \to 0 \), and the closed unit ball \( \{ x : \lVert x\rVert \le 1 \} \) is always convex, a consequence of the triangle inequality.<sup>[7](https://adwendland.github.io/applied-mathematics.pdf)</sup>

The reverse triangle inequality \( \bigl\lvert \lVert x\rVert - \lVert y\rVert \bigr\rvert \le \lVert x-y\rVert \) follows from the axioms and shows the norm is a continuous function on its own space.<sup>[5](https://courses.ms.ut.ee/2024/fa1/spring/uploads/Main/FA1--Lecture_Notes--2024-02-19.pdf)</sup> Not every metric on a vector space comes from a norm: the discrete metric is neither translation invariant nor homogeneous, so normed spaces form a strict subclass of metric spaces.<sup>[8](https://unseel.com/mathematics/metric-space)</sup>

## Examples

- **Euclidean space.** \( \mathbb{K}^n \) with the [Euclidean norm](https://www.edgepedia.org/euclidean-norm) \( \lVert x\rVert_2 = \bigl(\sum_{j=1}^{n} \lvert\xi_j\rvert^2\bigr)^{1/2} \) is the prototypical normed space.<sup>[5](https://courses.ms.ut.ee/2024/fa1/spring/uploads/Main/FA1--Lecture_Notes--2024-02-19.pdf)</sup> Other norms on the same underlying space include the supremum norm \( \lVert x\rVert_\infty = \max_i \lvert x_i\rvert \) and the sum norm \( \lVert x\rVert_1 = \sum_i \lvert x_i\rvert \); the same linear space with different norms counts as a different normed space.<sup>[7](https://adwendland.github.io/applied-mathematics.pdf)</sup>
- **Function spaces.** On \( C[a,b] \), the continuous real-valued functions on a compact interval, the supremum norm \( \lVert x\rVert_\infty = \max_{a\le t\le b} \lvert x(t)\rvert \) measures the largest deviation between functions.<sup>[7](https://adwendland.github.io/applied-mathematics.pdf)</sup> The Lebesgue spaces \( L^1 \), \( L^2 \), and \( L^\infty \) carry the norms \( \int_X \lvert f\rvert\,d\mu \), \( \bigl(\int_X \lvert f\rvert^2\,d\mu\bigr)^{1/2} \), and the essential supremum respectively.<sup>[6](https://www.math.stonybrook.edu/~aknapp/download/b2-realanal-inside.pdf)</sup>
- **Sequence spaces.** The space \( \ell^\infty \) of bounded sequences with \( \lVert x\rVert = \sup_n \lvert x_n\rvert \), and the \( \ell^p \) spaces, are standard infinite-dimensional examples.<sup>[7](https://adwendland.github.io/applied-mathematics.pdf)</sup>

## Completeness and Banach spaces

A normed space that is complete under the induced metric, meaning every [Cauchy sequence](https://www.edgepedia.org/cauchy-sequence) converges to a point of the space, is called a [Banach space](https://www.edgepedia.org/banach-space).<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup><sup> • </sup><sup>[4](https://mathshistory.st-andrews.ac.uk/Biographies/Banach/)</sup> Completeness is a property of the norm, not just of the underlying vector space: \( C[0,1] \) is complete under the supremum norm but not under the \( L^1 \) norm \( \int_0^1 \lvert f(t)\rvert\,dt \).<sup>[9](https://blogs.ethz.ch/kowalski/2009/02/12/banach-or-not-banach/)</sup> The polynomial space on \([0,1]\) is not complete under the supremum norm, since polynomials can converge uniformly to functions such as \( e^t \) that are not polynomials.<sup>[10](https://calc-path.com/real-analysis/modules/metric-spaces)</sup>

Every normed space has a Banach completion, a complete normed space containing the original as a dense subspace; any two completions are uniquely linearly isometric.<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup><sup> • </sup><sup>[11](https://alphabetamath.cc/item/ex-polynomials-are-not-complete-in-the-supremum-norm)</sup> Completeness matters because the central theorems of functional analysis, including the uniform boundedness principle, the open mapping theorem, and the closed graph theorem, require the domain to be complete.<sup>[12](https://www.terasystems.ai/tera-math-scholar/functional/ch1.html)</sup>

## Finite-dimensional normed spaces

Finite-dimensional normed spaces behave like [Euclidean space](https://www.edgepedia.org/euclidean-space) regardless of the norm chosen. Three results, proved from the equivalence of norms together with the completeness of \( \mathbb{R} \) and \( \mathbb{C} \), summarize the situation.<sup>[3](https://www.maths.ox.ac.uk/system/files/attachments/lecture-notes-MT-2018-Nov-24.pdf)</sup>

- **All norms are equivalent.** On a finite-dimensional space, any two norms \( \lVert\cdot\rVert \) and \( \lVert\cdot\rVert_0 \) admit constants \( C_1, C_2 \) with \( \lVert x\rVert \le C_1 \lVert x\rVert_0 \) and \( \lVert x\rVert_0 \le C_2 \lVert x\rVert \) for all \( x \), so they induce the same topology and the same convergent sequences.<sup>[3](https://www.maths.ox.ac.uk/system/files/attachments/lecture-notes-MT-2018-Nov-24.pdf)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup> This fails in infinite dimensions, where different norms on the same space can produce different completeness behavior and different topologies.<sup>[9](https://blogs.ethz.ch/kowalski/2009/02/12/banach-or-not-banach/)</sup>
- **Every linear map is bounded.** If \( X \) is finite dimensional and \( Y \) is any normed space, every linear map \( T : X \to Y \) is a bounded linear operator; the proof introduces the auxiliary norm \( \lVert x\rVert_T = \lVert x\rVert_X + \lVert Tx\rVert_Y \) and uses its equivalence with \( \lVert\cdot\rVert_X \).<sup>[3](https://www.maths.ox.ac.uk/system/files/attachments/lecture-notes-MT-2018-Nov-24.pdf)</sup>
- **Finite-dimensional spaces are Banach spaces.** Choosing a basis \( f_1, \ldots, f_m \) gives a linear bijection \( Q : \mathbb{F}^m \to X \), \( Q(\mu_1,\ldots,\mu_m) = \sum_{i=1}^m \mu_i f_i \), which is a homeomorphism; since \( \mathbb{F}^m \) with the Euclidean norm is complete, so is \( X \).<sup>[3](https://www.maths.ox.ac.uk/system/files/attachments/lecture-notes-MT-2018-Nov-24.pdf)</sup> A consequence is that every finite-dimensional subspace of a normed space is complete and therefore closed, while general subspaces need not be closed.<sup>[3](https://www.maths.ox.ac.uk/system/files/attachments/lecture-notes-MT-2018-Nov-24.pdf)</sup>

The equivalence-of-norms argument relies on the completeness of the scalar field.

## Relation to inner product spaces and Hilbert spaces

An inner product \( (\cdot,\cdot) \) on a vector space induces a norm by \( \lVert x\rVert = \sqrt{(x,x)} \), so every inner product space is a normed space, and a complete inner product space is a [Hilbert space](https://www.edgepedia.org/hilbert-space).<sup>[13](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/functional_analysis.html)</sup> The converse has a clean characterization: a norm is generated by an inner product if and only if it satisfies the parallelogram law \( \lVert x+y\rVert^2 + \lVert x-y\rVert^2 = 2(\lVert x\rVert^2 + \lVert y\rVert^2) \) for all \( x, y \).<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup><sup> • </sup><sup>[14](https://oshalit.net.technion.ac.il/2012/11/02/advanced-analysis-notes-6-banach-spaces-basics-the-hahn-banach-theorems/)</sup> The supremum norm on \( C[0,1] \) makes that space a Banach space, but the norm fails the parallelogram law, so it does not come from an inner product.<sup>[15](https://stationarystates.com/mathematical-physics/spaces-that-narrowly-miss-becoming-hilbert-spaces/)</sup>

## History

[Stefan Banach](https://www.edgepedia.org/stefan-banach) (1892–1945), in his 1920 dissertation written at Jan Kazimierz University in Lwów and published in 1922 in *Fundamenta Mathematicae*, defined axiomatically what today is called a Banach space, and he founded modern functional analysis by developing a systematic theory where earlier work by Volterra, Fredholm, Hilbert, and Riesz consisted of isolated results.<sup>[4](https://mathshistory.st-andrews.ac.uk/Biographies/Banach/)</sup><sup> • </sup><sup>[16](https://link.springer.com/article/10.1007/s13398-024-01636-6)</sup> The idea was introduced independently at about the same time; [Norbert Wiener](https://www.edgepedia.org/norbert-wiener) introduced the notion but did not develop the theory, and the name "Banach space" was coined by Maurice Fréchet.<sup>[4](https://mathshistory.st-andrews.ac.uk/Biographies/Banach/)</sup> In 1910 [Frigyes Riesz](https://www.edgepedia.org/frigyes-riesz) had already introduced the normed function space \( L^p \), and in 1922 Banach, Hans Hahn, and Norbert Wiener published papers using the same or similar sets of axioms.<sup>[17](https://www.sciencedirect.com/topics/mathematics/abstract-space)</sup><sup> • </sup><sup>[18](https://mathshistory.st-andrews.ac.uk/Miller/mathword/b/)</sup> Banach's 1932 monograph *Théorie des Opérations Linéaires* is regarded as the founding work of linear functional analysis.<sup>[16](https://link.springer.com/article/10.1007/s13398-024-01636-6)</sup> On the topological side, Andrey Kolmogorov showed in 1934 that a topological vector space is normable, meaning its topology comes from a norm, exactly when it has a convex bounded neighborhood of zero.<sup>[2](https://encyclopediaofmath.org/wiki/Norm)</sup>

## References

1. [Functional Analysis Lecture Notes, Spring 2020, MIT OpenCourseWare (18.102)](https://ocw.mit.edu/courses/18-102-introduction-to-functional-analysis-spring-2021/3d4cc88026d44a01f936cd6a0aa995cb_MIT18_102s20_lec_FA.pdf)
2. [Norm, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Norm)
3. [Finite Dimensional Normed Spaces, Oxford Mathematics lecture notes (MT 2018)](https://www.maths.ox.ac.uk/system/files/attachments/lecture-notes-MT-2018-Nov-24.pdf)
4. [Stefan Banach (1892–1945), Biography, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Banach/)
5. [Functional Analysis I, Lecture Notes, University of Tartu (2024)](https://courses.ms.ut.ee/2024/fa1/spring/uploads/Main/FA1--Lecture_Notes--2024-02-19.pdf)
6. [L1, L2, L∞, and Normed Linear Spaces, Stony Brook real analysis notes](https://www.math.stonybrook.edu/~aknapp/download/b2-realanal-inside.pdf)
7. [Normed Linear Spaces, Applied Mathematics notes, A. Dwendland](https://adwendland.github.io/applied-mathematics.pdf)
8. [Metric Space — Distance Without a Vector Space, Unseel](https://unseel.com/mathematics/metric-space)
9. [Banach or not Banach?, E. Kowalski's blog, ETH Zürich](https://blogs.ethz.ch/kowalski/2009/02/12/banach-or-not-banach/)
10. [Real Analysis — Metric Spaces & Topology, Calc-Path](https://calc-path.com/real-analysis/modules/metric-spaces)
11. [Polynomials are not complete in the supremum norm on a compact interval, Alphabeta Math](https://alphabetamath.cc/item/ex-polynomials-are-not-complete-in-the-supremum-norm)
12. [Chapter 1: Normed and Banach Spaces, Tera Math Scholar](https://www.terasystems.ai/tera-math-scholar/functional/ch1.html)
13. [Relevant concepts from functional analysis, MA8502, NTNU](https://www.math.ntnu.no/emner/MA8502/2024h/lectures/html/chapter_01/functional_analysis.html)
14. [Advanced Analysis, Notes 6: Banach spaces, Orr Moshe Shalit, Technion](https://oshalit.net.technion.ac.il/2012/11/02/advanced-analysis-notes-6-banach-spaces-basics-the-hahn-banach-theorems/)
15. [Spaces that narrowly miss becoming Hilbert Spaces, Stationary States](https://stationarystates.com/mathematical-physics/spaces-that-narrowly-miss-becoming-hilbert-spaces/)
16. [The Banach Fixed Point Theorem: selected topics from its hundred-year history, RACSAM (2024)](https://link.springer.com/article/10.1007/s13398-024-01636-6)
17. [Abstract Space, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/abstract-space)
18. [Earliest Known Uses of Some of the Words of Mathematics (B), MacTutor](https://mathshistory.st-andrews.ac.uk/Miller/mathword/b/)

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Analysis and mathematical models › Functional analysis*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license