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 $\mathbb {R} ^{\mathbb {N} }$) 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

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:

$${\begin{alignedat}{4}f:\;&&[0,1]&&\;\to \;&L^{2}(0,1)\\[0.3ex]&&t&&\;\mapsto \;&\mathbb {1} _{[0,t]}\\\end{alignedat}}$$

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 $X.$

Proposition — Given any two normalized crinkled arcs in a Hilbert space, each is unitarily equivalent to a reparameterization of the other.

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

• Differentiation in Fréchet spaces
• Differentiable vector–valued functions from Euclidean space – Differentiable function in functional analysis

References

• Einar Hille & Ralph Phillips: "Functional Analysis and Semi Groups", Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R.I., 1957.
• Halmos, Paul R. (8 November 1982). A Hilbert Space Problem Book. Graduate Texts in Mathematics. Vol. 19 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-90685-0. OCLC 8169781.