Infinite-dimensional vector function explained

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} for every positive integer k and every real number t. Then the function f defined by the formula$$f(t)\; =\; (f\_1(t),\; f\_2(t),\; f\_3(t),\; \backslash ldots)\backslash ,,$$takes values that lie in the infinite-dimensional vector space X (or \R ^{\N} ) of real-valued sequences. For example,$$f(2)\; =\; \backslash left(2,\; \backslash frac,\; \backslash frac,\; \backslash frac,\; \backslash frac,\; \backslash ldots\backslash right).$$

As a number of different topologies can be defined on the space

X, to talk about the derivative of f, it is first necessary to specify a topology on 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 (for example, the space of functions A \to 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.

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 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 ] \to X, where X is a Banach space or another topological vector space then the derivative of f can be defined in the usual way: $$f\text{'}(t)\; =\; \backslash lim\_\backslash frac.$$

Functions with values in a Hilbert space

If

f is a function of real numbers with values in a Hilbert space X, then the derivative of f at a point t can be defined as in the finite-dimensional case:$$f\text{'}(t)=\backslash lim\_\; \backslash frac.$$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 \in R ^{n} or even t\in Y, where Y is an infinite-dimensional vector space).

If

X is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if$$f\; =\; (f\_1,f\_2,f\_3,\backslash ldots)$$(that is, f = f _{1} e _{1+f} _{2} e _{2+f} _{3} e _{3+ … ,} where e _{1,e} _{2,e} _{3,\ldots} is an orthonormal basis of the space X ), and f'(t) exists, then$$f\text{'}(t)\; =\; (f\_1\text{'}(t),f\_2\text{'}(t),f\_3\text{'}(t),\backslash ldots).$$However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual topology of the Hilbert space.

Most of the above hold for other topological vector spaces

X too. However, not as many classical results hold in the Banach space setting, for example, an absolutely continuous function with values in a suitable Banach space need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.

Crinkled arcs

See main article: Crinkled arc.

If

[a, b] is an interval contained in the domain of a curve f that is valued in a topological vector space then the vector f(b) - f(a) is called the chord of f determined by [a, b] . If [c, d] is another interval in its domain then the two chords are said to be non−overlapping chords if [a, b] and [c, d] have at most one end−point in common. Intuitively, two non−overlapping chords of a curve valued in an inner product space are orthogonal vectors if the curve makes a right angle turn somewhere along its path between its starting point and its ending point. If every pair of non−overlapping chords are orthogonal then such a right turn happens at every point of the curve; such a curve can not be differentiable at any point. A crinkled arc is an injective continuous curve with the property that any two non−overlapping chords are orthogonal vectors. An example of a crinkled arc in the Hilbert L ^{2} space L ^{2(0,} 1 ) is:where 1 _{[0,t]} : ( 0 , 1 ) \to \{ 0 , 1 \} is the indicator function defined byA crinkled arc can be found in every infinite−dimensional Hilbert space because any such space contains a closed vector subspace that is isomorphic to L ^{2(0,} 1 ). A crinkled arc f : [ 0 , 1 ] \to X is said to be normalized if f( 0 ) = 0 , \|f( 1 )\| = 1 , and the span of its image f([ 0 , 1 ]) is a dense subset of X.

If

h : [ 0 , 1 ] \to [ 0 , 1 ] is an increasing homeomorphism then f \circ h is called a reparameterization of the curve f : [ 0 , 1 ] \to X. Two curves f and g in an inner product space X are unitarily equivalent if there exists a unitary operator L : X \to X (which is an isometric linear bijection) such that g = L \circ f (or equivalently, f = L ^{-1} \circ g ).

Measurability

The measurability of

f can be defined by a number of ways, most important of which are Bochner measurability and weak measurability.

Integrals

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} spaces have been defined for such functions.

References

• Einar Hille & Ralph Phillips: "Functional Analysis and Semi Groups", Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R.I., 1957.