1. Example

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

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

Moreover, for any set ${\displaystyle A,}$ there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of ${\displaystyle A}$ (for example, the space of functions ${\displaystyle A\to K}$ with finitely-many nonzero elements, where ${\displaystyle K}$ is the desired field of scalars). Furthermore, the argument ${\displaystyle t}$ 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, ${\displaystyle X}$ 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.

2.1. Derivatives

If ${\displaystyle f:[0,1]\to X,}$ where ${\displaystyle X}$ is a Banach space or another topological vector space then the derivative of ${\displaystyle f}$ can be defined in the usual way: ${\displaystyle f^{\prime }(t)=\underset{h\to 0}{lim}\frac{f(t+h)-f(t)}{h}.}$

Functions with values in a Hilbert space

If ${\displaystyle f}$ is a function of real numbers with values in a Hilbert space ${\displaystyle X,}$ then the derivative of ${\displaystyle f}$ at a point ${\displaystyle t}$ can be defined as in the finite-dimensional case: ${\displaystyle f^{\prime }(t)=\underset{h\to 0}{lim}\frac{f(t+h)-f(t)}{h}.}$ 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, ${\displaystyle t\in R^n}$ or even ${\displaystyle t\in Y,}$ where ${\displaystyle Y}$ is an infinite-dimensional vector space).

If ${\displaystyle X}$ is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if ${\displaystyle f=(f_1,f_2,f_3,\dots )}$ (that is, ${\displaystyle f=f_1{e}_1+f_2{e}_2+f_3{e}_3+\cdots ,}$ where ${\displaystyle e_1,e_2,e_3,\dots }$ is an orthonormal basis of the space ${\displaystyle X}$), and ${\displaystyle f^{\prime }(t)}$ exists, then ${\displaystyle f^{\prime }(t)=(f_1^{\prime }(t),f_2^{\prime }(t),f_3^{\prime }(t),\dots ).}$ 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 ${\displaystyle X}$ 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.

2.2. Crinkled Arcs

If ${\displaystyle [a,b]}$ is an interval contained in the domain of a curve ${\displaystyle f}$ that is valued in a topological vector space then the vector ${\displaystyle f(b)-f(a)}$ is called the chord of ${\displaystyle f}$ determined by ${\displaystyle [a,b]}$.^[1] If ${\displaystyle [c,d]}$ is another interval in its domain then the two chords are said to be non−overlapping chords if ${\displaystyle [a,b]}$ and ${\displaystyle [c,d]}$ 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 ${\displaystyle L^2}$ space ${\displaystyle L^2(0,1)}$ is:^[2] $\text{Unknown environment 'alignat'}$ where ${\displaystyle {\mathbb{1}}_{[0,t]}:(0,1)\to \{0,1\}}$ is the indicator function defined by $\text{Unknown environment 'cases'}$ 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 ${\displaystyle L^2(0,1).}$^[2] A crinkled arc ${\displaystyle f:[0,1]\to X}$ is said to be normalized if ${\displaystyle f(0)=0,}$ ${\displaystyle \Vert f(1)\Vert =1,}$ and the span of its image ${\displaystyle f([0,1])}$ is a dense subset of ${\displaystyle X.}$^[2]

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

2.3. Measurability

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