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A graph of the vector-valued function r(z) = 2 cos z, 4 sin z, z indicating a range of solutions and the vector when evaluated near z = 19.5 A common example of a vector-valued function is one that depends on a single real parameter t , often representing time , producing a vector v ( t ) as the result.
In a vector space, the null vector is the neutral element of vector addition; depending on the context, a null vector may also be a vector mapped to some null by a function under consideration (such as a quadratic form coming with the vector space, see null vector, a linear mapping given as matrix product or dot product, [4] a seminorm in a ...
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition and scalar multiplication.
In the theory of vector measures, Lyapunov's theorem states that the range of a finite-dimensional vector measure is closed and convex. [1] [2] [3] In fact, the range of a non-atomic vector measure is a zonoid (the closed and convex set that is the limit of a convergent sequence of zonotopes). [2]
The curl of a vector field F, denoted by curl F, or , or rot F, is an operator that maps C k functions in R 3 to C k−1 functions in R 3, and in particular, it maps continuously differentiable functions R 3 → R 3 to continuous functions R 3 → R 3. It can be defined in several ways, to be mentioned below:
Vector optimization is a subarea of mathematical optimization where optimization problems with a vector-valued objective functions are optimized with respect to a given partial ordering and subject to certain constraints.
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.