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When the scalar field is the real numbers, the vector space is called a real vector space, and when the scalar field is the complex numbers, the vector space is called a complex vector space. [4] These two cases are the most common ones, but vector spaces with scalars in an arbitrary field F are also commonly considered.
Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L.
Normed vector space, a vector space on which a norm is defined; Hilbert space; Ordered vector space, a vector space equipped with a partial order; Super vector space, name for a Z 2-graded vector space; Symplectic vector space, a vector space V equipped with a non-degenerate, skew-symmetric, bilinear form
This is a list of vector spaces in abstract mathematics, by Wikipedia page. Banach space; Besov space; Bochner space; Dual space; Euclidean space; Fock space; Fréchet space; Hardy space; Hilbert space; Hölder space; LF-space; L p space; Minkowski space; Montel space; Morrey–Campanato space; Orlicz space; Riesz space; Schwartz space; Sobolev ...
A diagram of dimensions 1, 2, 3, and 4. In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. [1] [2] It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension.
A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called unoriented. In mathematics , orientability is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left ...
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space [1] [2]) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar , often denoted with angle brackets such as in a , b {\displaystyle \langle a,b\rangle } .
Topological vector space This page was last edited on 5 March 2023, at 03:35 (UTC). Text is available under the Creative Commons Attribution-ShareAlike ...