Search results
Results from the WOW.Com Content Network
Most results of the finite-dimensional case also hold in the infinite-dimensional case too, with some modifications. Differentiation can also be defined to functions of several variables (for example, t ∈ R n {\displaystyle t\in R^{n}} or even t ∈ Y , {\displaystyle t\in Y,} where Y {\displaystyle Y} is an infinite-dimensional vector space).
The space of all functions from X to V is commonly denoted V X. If X is finite and V is finite-dimensional then V X has dimension |X|(dim V), otherwise the space is infinite-dimensional (uncountably so if X is infinite). Many of the vector spaces that arise in mathematics are subspaces of some function space. We give some further examples.
In mathematics, infinite-dimensional holomorphy is a branch of functional analysis. It is concerned with generalizations of the concept of holomorphic function to functions defined and taking values in complex Banach spaces (or Fréchet spaces more generally), typically of infinite dimension.
This space is the infinite-dimensional generalization of the space of finite-dimensional vectors. It is usually the first example used to show that in infinite-dimensional spaces, a set that is closed and bounded is not necessarily (sequentially) compact (as is the case in all finite dimensional spaces). Indeed, the set of orthonormal vectors ...
A theorem attributed to Mazur [7] asserts that every infinite-dimensional Banach space V contains a basic sequence, i.e., there is an infinite-dimensional subspace of V that has a Schauder basis. Examples
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 ...
Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the ...
The question of whether every infinite-dimensional Banach space contains an isomorph of some ℓ p or of c 0, was answered negatively by B. S. Tsirelson's construction of Tsirelson space in 1974. The dual statement, that every separable Banach space is linearly isometric to a quotient space of ℓ 1 , was answered in the affirmative by Banach ...