An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or a Banach space. Such functions are applied in most sciences including physics.
Example Set f k ( t ) = t / k 2 {\displaystyle f_{k}(t)=t/k^{2}} for every positive integer k {\displaystyle k} and every real number t . {\displaystyle t.} Then the function f {\displaystyle f} defined by the formula
f ( t ) = ( f 1 ( t ) , f 2 ( t ) , f 3 ( t ) , … ) , {\displaystyle f(t)=(f_{1}(t),f_{2}(t),f_{3}(t),\ldots )\,,}
takes values that lie in the infinite-dimensional vector space X {\displaystyle X} (or R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} ) of real-valued sequences. For example,
f ( 2 ) = ( 2 , 2 4 , 2 9 , 2 16 , 2 25 , … ) . {\displaystyle f(2)=\left(2,{\frac {2}{4}},{\frac {2}{9}},{\frac {2}{16}},{\frac {2}{25}},\ldots \right).}
As a number of different topologies can be defined on the space X , {\displaystyle X,} to talk about the derivative of f , {\displaystyle f,} it is first necessary to specify a topology on X {\displaystyle X} or the concept of a limit in X . {\displaystyle X.}
Moreover, for any set A , {\displaystyle A,} there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of A {\displaystyle A} (for example, the space of functions A → K {\displaystyle A\to K} with finitely-many nonzero elements, where K {\displaystyle K} is the desired field of scalars). Furthermore, the argument t {\displaystyle t} could lie in any set instead of the set of real numbers.
Integral and derivative Most theorems on integration and differentiation of scalar functions can be generalized to vector-valued functions, often using essentially the same proofs. Perhaps the most important exception is that absolutely continuous functions need not equal the integrals of their (a.e.) derivatives (unless, for example, X {\displaystyle X} is a Hilbert space); see Radon–Nikodym theorem A curve is a continuous map of the unit interval (or more generally, of a non−degenerate closed interval of real numbers) into a topological space. An arc is a curve that is also a topological embedding. A curve valued in a Hausdorff space is an arc if and only if it is injective.
Derivatives If f : [ 0 , 1 ] → X , {\displaystyle f:[0,1]\to X,} where X {\displaystyle X} is a Banach space or another topological vector space then the derivative of f {\displaystyle f} can be defined in the usual way:
f ′ ( t ) = lim h → 0 f ( t + h ) − f ( t ) h . {\displaystyle f'(t)=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}.}
Functions with values in a Hilbert space If f {\displaystyle f} is a function of real numbers with values in a Hilbert space X , {\displaystyle X,} then the derivative of f {\displaystyle f} at a point t {\displaystyle t} can be defined as in the finite-dimensional case:
f ′ ( t ) = lim h → 0 f ( t + h ) − f ( t ) h . {\displaystyle f'(t)=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}.}
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