Normed vector spaces

This page is a sub-page of our page on Vector spaces.

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The sub-pages of this page are:

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Related KMR-pages:

Topology
Topological vector spaces
Vector spaces of finite and infinite dimensions

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Other relevant sources of information:

Euclidean spaces
Metric spaces
Inner product spaces

Vector spaces
Normed vector spaces
Normed vector spaces and inner product spaces
Banach spaces
The p-norm in finite dimensions
{\ell}^p -spaces
L^p -spaces
L^{\infty} -spaces
Hilbert spaces

Sequence spaces

Orthonormal basis
Dual basis

Fourier series
Fourier transform

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Spaces of countably infinite dimensions
(Sequence spaces and spaces of periodic functions)

Quoting Wikipedia (on Sequence spaces):

In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K, and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space.

The most important sequence spaces in analysis are the \, ℓ^p spaces, consisting of the p-power summable sequences, with the p-norm. These are special cases of \, L^p \, spaces for the counting measure on the set of natural numbers. Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted \, c \, and \, c_0 , with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence, under which it becomes a special kind of Fréchet space called FK-space.

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26 types of mathematical spaces:
26 types of mathematical spaces

Spaces of uncountably infinite dimensions
(Spaces of non-periodic functions)

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