Infinite-dimensional vector function

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 $$f(t) = (f_1(t), f_2(t), f_3(t), \ldots)\, ,$$ takes values that lie in the infinite-dimensional vector space $$X$$ (or $$\R^{\N}$$) of real-valued sequences. For example, $$f(2) = \left(2, \frac{2}{4}, \frac{2}{9}, \frac{2}{16}, \frac{2}{25}, \ldots\right).$$

As a number of different topologies can be defined on the space $$X,$$ 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: $$f'(t) = \lim_{h\to 0}\frac{f(t+h)-f(t)}{h}.$$

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: $$f'(t)=\lim_{h\to 0} \frac{f(t+h)-f(t)}{h}.$$ 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 $$f = (f_1,f_2,f_3,\ldots)$$ (that is, $$f = f_1 e_1+f_2 e_2+f_3 e_3+\cdots,$$ where $$e_1,e_2,e_3,\ldots$$ is an orthonormal basis of the space $$X$$), and $$f'(t)$$ exists, then $$f'(t) = (f_1'(t),f_2'(t),f_3'(t),\ldots).$$ 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
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: $$\begin{alignat}{4} f :\;&& [0, 1] &&\;\to   \;& L^2(0, 1) \\[0.3ex] && t     &&\;\mapsto\;& \mathbb{1}_{[0,t]} \\ \end{alignat}$$ where $$\mathbb{1}_{[0,\,t]} : (0, 1) \to \{0, 1\}$$ is the indicator function defined by $$x \;\mapsto\; \begin{cases}1 & \text{ if } x \in [0, t]\\ 0 & \text{ otherwise } \end{cases}$$ A 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 $$X.$$

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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.