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In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure , which takes nonnegative real values only.
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 ...
function greater_h (kernel h, int p, int q, real u): // h is the kernel function, h(i,j) gives the ith, jth entry of H // p and q are the number of rows and columns of the kernel matrix H // vector of size p P: = vector (p) // indexing from zero j: = 0 // starting from the bottom, compute the [[supremum|least upper bound]] for each row for i ...
Vectors are defined in spherical coordinates by (r, θ, φ), where r is the length of the vector, θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and; φ is the angle between the projection of the vector onto the xy-plane and the positive X-axis (0 ≤ φ < 2π).
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.
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.
In abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map: that is separately -linear in each of its arguments. [1] More generally, one can define multilinear forms on a module over a commutative ring.