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An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an abelian group under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a ring homomorphism from the ...
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
More generally, the Kuratowski embedding allows one to see any metric space as a subspace of a normed vector space. Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in functional analysis. Completeness is particularly important in this context: a complete normed vector space is known as a Banach space.
In viewing a tensor as a multilinear map, it is conventional to identify the double dual V ∗∗ of the vector space V, i.e., the space of linear functionals on the dual vector space V ∗, with the vector space V. There is always a natural linear map from V to its double dual, given by evaluating a linear form in V ∗ against a vector in V.
A scalar in physics and other areas of science is also a scalar in mathematics, as an element of a mathematical field used to define a vector space.For example, the magnitude (or length) of an electric field vector is calculated as the square root of its absolute square (the inner product of the electric field with itself); so, the inner product's result is an element of the mathematical field ...
A scalar is an element of a field which is used to define a vector space.In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector.
A subset of a vector space over an ordered field is a cone (or sometimes called a linear cone) if for each in and positive scalar in , the product is in . [2] Note that some authors define cone with the scalar ranging over all non-negative scalars (rather than all positive scalars, which does not include 0). [3]