Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Normed vector space
Summary
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 a generalization of the intuitive notion of "length" in the physical world. If is a vector space over , where is a field equal to or to , then a norm on is a map , typically denoted by , satisfying the following four axioms:
Non-negativity: for every ,.
Positive definiteness: for every , if and only if is the zero vector.
Absolute homogeneity: for every and ,
Triangle inequality: for every and ,
If is a real or complex vector space as above, and is a norm on , then the ordered pair is called a normed vector space. If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by .
A norm induces a distance, called its , 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, a major subfield of mathematics.
Seminormed space
A normed vector space is a vector space equipped with a norm. A is a vector space equipped with a seminorm.
A useful variation of the triangle inequality is
for any vectors and
This also shows that a vector norm is a (uniformly) continuous function.
Property 3 depends on a choice of norm on the field of scalars.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related courses (40)
MATH-305: Introduction to partial differential equations
This is an introductory course on Elliptic Partial Differential Equations. The course will cover the theory of both classical and generalized (weak) solutions of elliptic PDEs.
MATH-111(e): Linear Algebra
L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.
COM-406: Foundations of Data Science
We discuss a set of topics that are important for the understanding of modern data science but that are typically not taught in an introductory ML course. In particular we discuss fundamental ideas an