Infinite-dimensional vector function

Infinite-dimensional vector function refers to a function whose values lie in an infinite-dimensional vector space, such as a Hilbert space or a Banach space. Infinite-dimensional vector function refers to a function whose values lie in an infinite-dimensional vector space, such as a Hilbert space or a Banach space. Such functions are applied in most sciences including physics. Set f k ( t ) = t / k 2 {displaystyle f_{k}(t)=t/k^{2}} for every positive integer k and every real number t. Then values of the function lie in the infinite-dimensional vector space X (or R N {displaystyle mathbf {R} ^{mathbf {N} }} ) of real-valued sequences. For example, As a number of different topologies can be defined on the space X, we cannot talk about the derivative of f without first defining the topology of X or the concept of a limit in X. Moreover, for any set A, there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of A (e.g., the space of functions A → K {displaystyle A ightarrow K} with finitely-many nonzero elements, where K is the desired field of scalars). Furthermore, the argument t could lie in any set instead of the set of real numbers. If, e.g., f : [ 0 , 1 ] → X {displaystyle f: ightarrow X} , where X is a Banach space or another topological vector space, the derivative of f can be defined in the standard way: f ′ ( t ) := lim h → 0 f ( t + h ) − f ( t ) h {displaystyle f'(t):=lim _{h ightarrow 0}{frac {f(t+h)-f(t)}{h}}} . The measurability of f can be defined by a number of ways, most important of which are Bochner measurability and weak measurability. The most important integrals of f are called Bochner integral (when X is a Banach space) and Pettis integral (when X is a topological vector space). Both these integrals commute with linear functionals. Also L p {displaystyle L^{p}} spaces have been defined for such functions.