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A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the ...
In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms.
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.
In mathematical analysis, the spaces of test functions and distributions are topological vector spaces (TVSs) that are used in the definition and application of distributions. Test functions are usually infinitely differentiable complex -valued (or sometimes real -valued) functions on a non-empty open subset U ⊆ R n {\displaystyle U\subseteq ...
A continuous function : from a non-empty and non-degenerate interval into a topological space is called a curve or a curve in . A path in X {\displaystyle X} is a curve in X {\displaystyle X} whose domain is compact while an arc or C 0 -arc in X {\displaystyle X} is a path in X {\displaystyle X} that is also a topological embedding .
Again take the field to be R, but now let the vector space V be the set R R of all functions from R to R. Let C(R) be the subset consisting of continuous functions. Then C(R) is a subspace of R R. Proof: We know from calculus that 0 ∈ C(R) ⊂ R R. We know from calculus that the sum of continuous functions is continuous.
One may define additive functions with values in any additive monoid (for example any group or more commonly a vector space). For sigma-additivity, one needs in addition that the concept of limit of a sequence be defined on that set. For example, spectral measures are sigma-additive functions with values in a Banach algebra.
The term "Helmholtz theorem" can also refer to the following. Let C be a solenoidal vector field and d a scalar field on R 3 which are sufficiently smooth and which vanish faster than 1/r 2 at infinity. Then there exists a vector field F such that = =; if additionally the vector field F vanishes as r → ∞, then F is unique. [18]