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.

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

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