Topological vector space (7707 bytes)
1: ...ausdorff space|Hausdorff]] [[topology]] such that vector addition ''X'' × ''X'' → ''X'' and sca...
3: ...al analysis]]. The elements of topological vector spaces are typically [[function (mathematics)|function]...
7: ...bert space]]s) are examples of topological vector spaces.
9: ...t ''f''(''x'') for every real number ''x''. This space is complete, but not normable.
11: Spaces of differentiable functions give other non-norma...

Locally convex topological vector space (1481 bytes)
1: ...n [[functional analysis]], a [[topological vector space]] is called
3: ...d (topology)|neighborhood]]s of 0. Every [[normed space]] is locally convex, since the [[triangle inequal...
5: ...s. Although such a space need not be [[Hausdorff_space|Hausdorff]], this is often also assumed.
7: ...spaces generalises parts of the theory of Banach spaces. Indeed, local convexity is a generalisation of...
9: ... On the other hand, [[Lp space|L^''p'' spaces]] for <math>0<p<1</math> are not locally convex.

Null vector (vector space) (207 bytes)
1: ... written 0, that is the [[neutral element]] for [[vector addition]].

Topological space (19036 bytes)
1: ...he branch of mathematics that studies topological spaces in their own right is called [[topology]].
5: A '''topological space''' is a [[set]] ''X'' together with a collection ...
15: ...be placed on nearly any set to form a topological space. When every set in a topology ''T''₁ i...
19: ...m. From the standpoint of topology, homeomorphic spaces are essentially identical.
21: ...topological spaces]], '''Top''', with topological spaces as [[object (category theory)|objects]] and cont...

Vector space example 1 (5154 bytes)
3: <br>Define Vector Addition:
7: Then '''V''' is a Vector Space over '''R'''.
46: <br>This vector space is denoted <b>Rⁿ</b>.
50: :''See also :'' [[Vector space]]

Vector space (7945 bytes)
1: ...ra]] is that of a '''vector space''' or '''linear space'''.
3: ...f a mathematical structure which we call a vector space.
5: ...''n'' with real-valued coefficients form a vector space, for example. It is this abstract quality that ma...
9: A [[set]] V is a '''vector space''' over a [[field (mathematics)|field]] F (for ex...
10: * an operation ''vector addition'' defined in V, denoted '''v''' + '''w''...

Vector space example 2 (2549 bytes)
13: Define vector addition in '''M''':
53: Then '''M''' is a vector space over '''C''' and we denote this as '''C'''<sup>m&...
57: ...e. In analysis, a vector space is called a linear space.
59: :''See also: [[Vector space]]''

Vector space (example 3) (2526 bytes)
1: ...ften called ''linear spaces'' instead of ''vector spaces''. This third example is one such set of functio...
6: ...osed interval]] [a,b] <tt>-></tt> '''R'''. Define vector addition:
10: Then C[a,b] is a vector space over the field '''R'''.

Normed vector space (4623 bytes)
1: ...>. It turns out that the following properties of "vector length" are the crucial ones.
3: ...y positive length. The only exception is the zero vector which has length zero.
4: # multiplying a vector by a positive number has the same effect on the l...
7: ...norm is defined is then called a '''normed vector space'''.
11: ...air (''V'',||&middot||) where ''V'' is a [[vector space]] and ||&middot|| a [[norm (mathematics)|norm]] o...

Vector space model (555 bytes)
1: ...nner by the use of vectors in a multi-dimensional space. It has been used first time by the [[SMART Infor...
3: Models basing on and extending the Vector Space Model:
4: * [[Generalized Vector Space Model]]
5: * [[Topic-based Vector Space Model]]

Space vector inducer (1613 bytes)
8: ...alactic centers]]. It is currently being used by space agencies such as [[NASA]] and the [[ESA]] to faci...

Symplectic vector space (3191 bytes)
1: ...]], a '''symplectic vector space''' is a [[vector space]] ''V'' equipped with a [[nondegenerate]], [[skew...
10: ==Standard symplectic space==
12: The standard symplectic space is '''R'''^2''n'' with the symplectic f...
14: ...mes; ''n'' [[identity matrix]]. In terms of basis vectors <math>(x_1, \ldots, x_n, y_1, \ldots, y_n)</math...
24: ==Subspaces==

Dimension (vector space) (3456 bytes)
1: ...e is uniquely defined. The dimension of the vectorspace ''V'' over the [[field (mathematics)|field]] ''F'...
7: E.g. The vector space '''R'''³ has {(1,0,0), (0,1,0), (0,0,1...
9: The [[complex number]]s '''C''' are a real vector space; we have dim_'''R'''('''C''') = 2 and d...
11: ... vector space with dimension 0 is {0}, the vector space consisting only of its zero element.
15: If ''W'' is a [[linear subspace]] of ''V'', then dim(''W'') &le; dim(''V'').

Graded vector space (1618 bytes)
1: ...tics]], a '''graded vector space''' is a [[vector space]] with an extra piece of structure, known as a ''...
3: == Graded vector spaces ==
5: A '''graded vector space''' is a vector space ''V'' which can be written as a [[direct sum]] of...
9: ...olynomial]]s in one variable form a graded vector space, where the homogeneous elements of degree ''n'' a...
11: == ''I''-graded vector spaces ==

Complex conjugate vector space (759 bytes)
1: ...itive and call it the '''complex conjugate vector space''' of V, V*. This comes with a canonical bijectiv...

Null vector (Minkowski space) (123 bytes)
1: ..., a '''null vector''' of [[Minkowski space]] is a vector with zero [[norm]].

Topic-based Vector Space Model (410 bytes)
1: ...en terms. A description of the Topic-based Vector Space Model can be found at [http://www.kuropka.net/Dat...

Soliton (topological) (1115 bytes)
1: A '''topological [[soliton]]''' is a solution of a system of [[par...
3: ...cording to Kibble-Higgs mechanism. The well known topological defects are [[magnetic monopoles]], [[cosmic stri...

Topological property (584 bytes)
1: ...cal invariant''' is a property of a [[topological space]] which is [[invariant_(mathematics)|invariant]] ...
3: ==Topological properties==
8: *[[compact space|compactness]]
10: *[[Hausdorff space|Hausdorff]]

Topological group (6068 bytes)
1: ...require that the topology on ''G'' be [[Hausdorff space|Hausdorff]]. The reasons, and some equivalent con...
3: ...tigated in [[mathematical analysis|analysis]] are topological groups (usually with some additional structure).
7: ...[[discrete group]]s. In this sense, the theory of topological groups subsumes that of ordinary groups.
9: ...h as [[Banach space]]s or [[Hilbert space]]s, are topological groups.
11: ... as a [[subspace (topology)|subset]] of Euclidean space '''R'''^{''n''&times;''n''}.

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