Normed Vector Spaces — Mathematics Knowledge System

Overview

A normed vector space is a vector space equipped with a norm — a function that assigns a non-negative real length to every vector. Norms generalise the familiar Euclidean distance to abstract vector spaces such as spaces of continuous functions, sequences, or integrable functions. They provide the geometric and analytic backbone of functional analysis: once a space has a norm, we can talk about convergence, continuity, and approximation.

Intuition

A norm is simply a consistent way to measure 'size' or 'length' for vectors in an abstract space. On the real line, the norm of a number is its absolute value. In the plane, the Euclidean norm is the familiar distance from the origin. The key axioms — positivity, scaling, and the triangle inequality — capture exactly what we need to do analysis: compare sizes, and know that the shortest path between two points is a straight line.

Formal Definition

Definition

Let X be a vector space over the field F (either R or C). A norm on X is a function ||·|| : X -> [0, infinity) satisfying three axioms: (1) positive definiteness: ||x|| = 0 if and only if x = 0; (2) absolute homogeneity: ||alpha x|| = |alpha| ||x|| for all scalars alpha; (3) triangle inequality: ||x + y|| <= ||x|| + ||y||. The pair (X, ||·||) is called a normed vector space. Every norm induces a metric d(x,y) = ||x - y||.

\|\alpha x\| = |\alpha|\,\|x\|

Scaling a vector scales its norm

Homogeneity

\|x + y\| \le \|x\| + \|y\|

Triangle inequality

d(x,y) = \|x - y\|

Induced metric

\|x\|_p = \left(\sum_{k=1}^n |x_k|^p\right)^{1/p}, \quad 1 \le p < \infty

p-norm on R^n

\|x\|_\infty = \max_{1 \le k \le n} |x_k|

Infinity norm on R^n

\|f\|_{C[a,b]} = \max_{t \in [a,b]} |f(t)|

Sup-norm on C[a,b]

Notation

Notation	Meaning
$(X, \\|\cdot\\|)$	Normed vector space X with norm \|\|·\|\|
$B(x,r) = \{y \in X : \\|y-x\\| < r\}$	Open ball of radius r centred at x
$\\|\cdot\\|_p$	p-norm (1 <= p <= infinity)
$\\|\cdot\\|_\infty$	Supremum norm

Properties

Reverse triangle inequality

\bigl|\|x\| - \|y\|\bigr| \le \|x - y\| \quad \text{for all } x,y \in X

Continuity of the norm

The map x \mapsto \|x\| is continuous from (X, \|\cdot\|) to [0,\infty).

Theorems

Theorem 1.1: Equivalence of Norms in Finite Dimensions

Any two norms on a finite-dimensional vector space X are equivalent: there exist constants c, C > 0 such that c||x||_a <= ||x||_b <= C||x||_a for all x. All norms on a finite-dimensional space induce the same topology.

Theorem 1.2: Riesz's Lemma

Let Y be a proper closed subspace of a normed space X and theta in (0,1). Then there exists x_theta in X with ||x_theta|| = 1 such that ||x_theta - y|| >= theta for all y in Y.

Theorem 1.3: Compactness of the Unit Ball

The closed unit ball of a normed space X is compact if and only if X is finite-dimensional.

Worked Examples

1
Let f, g be in C[0,1]. For any t in [0,1] we have |f(t) + g(t)| <= |f(t)| + |g(t)|.
2
Taking the maximum over t: max_t |f(t)+g(t)| <= max_t (|f(t)|+|g(t)|) <= max_t|f(t)| + max_t|g(t)|.
$\|f+g\|_\infty \le \|f\|_\infty + \|g\|_\infty$
3
Positive definiteness and homogeneity are immediate from properties of the absolute value. Hence ||·||_inf is a norm.

✓ Answer

The triangle inequality follows by taking the max after applying pointwise |f+g| <= |f|+|g|.

Practice Problems

Mediumproof writing

Prove that the norm function x |-> ||x|| is a Lipschitz map from (X, ||·||) to R, with Lipschitz constant 1.

EasyMultiple choice

Which of the following is NOT a norm on R^2?

||( x_1, x_2 )||_1 = |x_1| + |x_2|||( x_1, x_2 )|| = |x_1| + 2|x_2|||( x_1, x_2 )|| = |x_1 - x_2|||( x_1, x_2 )||_2 = sqrt(x_1^2 + x_2^2)

Mediumfree response

Give an example showing that in an infinite-dimensional normed space, equivalence of norms can fail. That is, find two non-equivalent norms on C[0,1].

Common Mistakes

Common Mistake

Confusing a norm with a metric: assuming every metric comes from a norm.

Not every metric is induced by a norm. For example, the discrete metric d(x,y)=1 for x!=y is not a norm metric. A metric comes from a norm only if it is translation-invariant and positively homogeneous.

Common Mistake

Believing all norms on infinite-dimensional spaces are equivalent.

Equivalence of norms holds only in finite dimensions. In infinite-dimensional spaces, different norms can give genuinely different topologies.

Quiz

Historical Background

The concept of a norm emerged gradually in the early twentieth century as mathematicians sought to extend calculus to infinite-dimensional settings. Stefan Banach's 1920 doctoral thesis, published in 1922, gave the first systematic treatment of complete normed spaces (now called Banach spaces). Earlier work by Fréchet (1906) on abstract metric spaces, and by Hilbert on his space of square-summable sequences, provided the building blocks. The axiomatic definition of a normed space crystallised by the 1930s through the combined efforts of Banach, Hahn, and Wiener.

1906
Fréchet introduces abstract metric spaces
Maurice Fréchet
1910
Hilbert studies l^2 and integral operators
David Hilbert
1922
Banach publishes his thesis, founding the theory of complete normed spaces
Stefan Banach
1932
Banach's monograph Théorie des opérations linéaires systematises the field
Stefan Banach

Summary

A normed space is a vector space with a consistent notion of length satisfying positivity, homogeneity, and the triangle inequality.
Every norm induces a metric, making any normed space a metric space.
In finite dimensions, all norms are equivalent (induce the same open sets and convergent sequences).
In infinite dimensions, norms can be inequivalent, and the closed unit ball is never compact.
Key examples: R^n with the p-norms, C[a,b] with the sup-norm, and sequence spaces l^p.

References

Websiteen.wikipedia.org
BookConway, J. — A Course in Functional Analysis, Chapter 1
Websiteen.wikipedia.org