Function Spaces
- See main article at Function space, especially the functional analysis section.
Let X be an arbitrary set and V an arbitrary vector space over F. The space of all functions from X to V is a vector space over F under pointwise addition and multiplication. That is, let f : X → V and g : X → V denote two functions, and let α∈F. We define
where the operations on the right hand side are those in V. The zero vector is given by the constant function sending everything to the zero vector in V. The space of all functions from X to V is commonly denoted VX.
If X is finite and V is finite-dimensional then VX 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.
Read more about this topic: Examples Of Vector Spaces
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