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Real Norm
Needed by:
Affine MMSE Estimators
Bounded Functions
Continuous Linear Transformations
Convex Functions
Distance Covariance Functions
Eigenvalues and Eigenvectors
Functional Analysis
Least Squares Linear Regressors
Minimum Mean Squared Error Estimates
Minimum Mean Squared Error Estimator
Norm Metrics
Real Inner Product Norms
Sequence Spaces
Supremum Norm
Topological Vector Spaces
Total Variation
Weighted Norms
Sheet PDF
Graph PDF



We generalize the concept of norm from real vectors to abstract vector spaces.


A norm is a real-valued functional that is (a) non-negative, (b) definite, (c) absolutely homogeneous, (d) and satisifies a triangle inequality. The triangle inequality property requires that the norm applied to the sum of any two vectors is less than the sum of the norms on those vectors.

A normed space (or norm space) is an ordered pair: a vector space whose field is the real or complex numbers and a norm on the space. We require the vector space to be over the field of real or complex numbers because of absolute homogeneity: the absolute value of a scalar must be defined.


Let $(X, \F )$ be a vector space where $\F $ is the field of real numbers or the field of complex numbers. Let $f: X \to \R $. The functional $f$ is a norm if

  1. $f(v) \geq 0$ for all $x \in V$
  2. $f(v) = 0$ if and only if $x = 0 \in X$.
  3. $f(\alpha x) = \abs{\alpha }f(x)$ for all $\alpha \in \F $, $x \in X$
  4. $f(x + y) \leq f(x) + f(y)$ for all $x, y \in X$.

In this case, for $x \in X$, we denote $f(x)$ by $\norm{x}$, read aloud “norm x”. The notation follows the notation for the absolute value function is a norm on the vector space of real numbers. In some cases, we go further, and for a norm indexed by some parameter $\alpha $ or set $A$ we write $\norm{x}_\alpha $ or $\norm{x}_A$.

When the field is assumed or clear from context, it is succinct to say let $(V, \norm{\cdot })$ be a normed space.


The absolute value function is a norm on the vector space of real numbers. In addition, the (Euclidean norm) is a norm on the vector space $\R ^n$.

Other terminology

The descriptive but slight more verbose norm vector space and normed vector space are also in usage.

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