1. Example

Set [math]\displaystyle{ f_k(t) = t/k^2 }[/math] for every positive integer [math]\displaystyle{ k }[/math] and every real number [math]\displaystyle{ t. }[/math] Then the function [math]\displaystyle{ f }[/math] defined by the formula [math]\displaystyle{ f(t) = (f_1(t), f_2(t), f_3(t), \ldots)\, , }[/math] takes values that lie in the infinite-dimensional vector space [math]\displaystyle{ X }[/math] (or [math]\displaystyle{ \R^{\N} }[/math]) of real-valued sequences. For example, [math]\displaystyle{ f(2) = \left(2, \frac{2}{4}, \frac{2}{9}, \frac{2}{16}, \frac{2}{25}, \ldots\right). }[/math]

As a number of different topologies can be defined on the space [math]\displaystyle{ X, }[/math] to talk about the derivative of [math]\displaystyle{ f, }[/math] it is first necessary to specify a topology on [math]\displaystyle{ X }[/math] or the concept of a limit in [math]\displaystyle{ X. }[/math]

Moreover, for any set [math]\displaystyle{ A, }[/math] there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of [math]\displaystyle{ A }[/math] (for example, the space of functions [math]\displaystyle{ A \to K }[/math] with finitely-many nonzero elements, where [math]\displaystyle{ K }[/math] is the desired field of scalars). Furthermore, the argument [math]\displaystyle{ t }[/math] could lie in any set instead of the set of real numbers.

2. 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, [math]\displaystyle{ X }[/math] 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.

2.1. Derivatives

If [math]\displaystyle{ f : [0,1] \to X, }[/math] where [math]\displaystyle{ X }[/math] is a Banach space or another topological vector space then the derivative of [math]\displaystyle{ f }[/math] can be defined in the usual way: [math]\displaystyle{ f'(t) = \lim_{h\to 0}\frac{f(t+h)-f(t)}{h}. }[/math]

Functions with values in a Hilbert space

If [math]\displaystyle{ f }[/math] is a function of real numbers with values in a Hilbert space [math]\displaystyle{ X, }[/math] then the derivative of [math]\displaystyle{ f }[/math] at a point [math]\displaystyle{ t }[/math] can be defined as in the finite-dimensional case: [math]\displaystyle{ f'(t)=\lim_{h\to 0} \frac{f(t+h)-f(t)}{h}. }[/math] 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, [math]\displaystyle{ t \in R^n }[/math] or even [math]\displaystyle{ t\in Y, }[/math] where [math]\displaystyle{ Y }[/math] is an infinite-dimensional vector space).

If [math]\displaystyle{ X }[/math] is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if [math]\displaystyle{ f = (f_1,f_2,f_3,\ldots) }[/math] (that is, [math]\displaystyle{ f = f_1 e_1+f_2 e_2+f_3 e_3+\cdots, }[/math] where [math]\displaystyle{ e_1,e_2,e_3,\ldots }[/math] is an orthonormal basis of the space [math]\displaystyle{ X }[/math]), and [math]\displaystyle{ f'(t) }[/math] exists, then [math]\displaystyle{ f'(t) = (f_1'(t),f_2'(t),f_3'(t),\ldots). }[/math] 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 [math]\displaystyle{ X }[/math] 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.

2.2. Crinkled Arcs

If [math]\displaystyle{ [a, b] }[/math] is an interval contained in the domain of a curve [math]\displaystyle{ f }[/math] that is valued in a topological vector space then the vector [math]\displaystyle{ f(b) - f(a) }[/math] is called the chord of [math]\displaystyle{ f }[/math] determined by [math]\displaystyle{ [a, b] }[/math].^[1] If [math]\displaystyle{ [c, d] }[/math] is another interval in its domain then the two chords are said to be non−overlapping chords if [math]\displaystyle{ [a, b] }[/math] and [math]\displaystyle{ [c, d] }[/math] have at most one end−point in common.^[1] 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.^[1] 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 [math]\displaystyle{ L^2 }[/math] space [math]\displaystyle{ L^2(0, 1) }[/math] is:^[2] [math]\displaystyle{ \begin{alignat}{4} f :\;&& [0, 1] &&\;\to \;& L^2(0, 1) \\[0.3ex] && t &&\;\mapsto\;& \mathbb{1}_{[0,t]} \\ \end{alignat} }[/math] where [math]\displaystyle{ \mathbb{1}_{[0,\,t]} : (0, 1) \to \{0, 1\} }[/math] is the indicator function defined by [math]\displaystyle{ x \;\mapsto\; \begin{cases}1 & \text{ if } x \in [0, t]\\ 0 & \text{ otherwise } \end{cases} }[/math] 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 [math]\displaystyle{ L^2(0, 1). }[/math]^[2] A crinkled arc [math]\displaystyle{ f : [0, 1] \to X }[/math] is said to be normalized if [math]\displaystyle{ f(0) = 0, }[/math] [math]\displaystyle{ \|f(1)\| = 1, }[/math] and the span of its image [math]\displaystyle{ f([0, 1]) }[/math] is a dense subset of [math]\displaystyle{ X. }[/math]^[2]

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

2.3. Measurability

The measurability of [math]\displaystyle{ f }[/math] can be defined by a number of ways, most important of which are Bochner measurability and weak measurability.

2.4. Integrals

The most important integrals of [math]\displaystyle{ f }[/math] are called Bochner integral (when [math]\displaystyle{ X }[/math] is a Banach space) and Pettis integral (when [math]\displaystyle{ X }[/math] is a topological vector space). Both these integrals commute with linear functionals. Also [math]\displaystyle{ L^p }[/math] spaces have been defined for such functions.