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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.
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.
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 ...
Pages in category "Vector spaces" ... Dimension (vector space) Examples of vector spaces; FinVect; Primordial element (algebra) Vector space; A. Anyonic Lie algebra; C.
This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic). A vector space is finite-dimensional if its dimension is a natural number.
The earliest examples of these were function spaces, each one adapted to its own class of problems. These examples shared many common features, and these features were soon abstracted into Hilbert spaces, Banach spaces, and more general topological vector spaces. These were a powerful toolkit for the solution of a wide range of mathematical ...
This more general type of spatial vector is the subject of vector spaces (for free vectors) and affine spaces (for bound vectors, as each represented by an ordered pair of "points"). One physical example comes from thermodynamics , where many quantities of interest can be considered vectors in a space with no notion of length or angle.
For example, if V and also X itself are vector spaces over F, the set of linear maps X → V form a vector space over F with pointwise operations (often denoted Hom(X,V)). One such space is the dual space of X : the set of linear functionals X → F with addition and scalar multiplication defined pointwise.