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

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

Proposition^[2] — 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 $f:[0,1]\to X.$ ^[1] 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.

References

^ ^a ^b ^c ^d Halmos 1982, pp. 5−7.
^ ^a ^b ^c ^d Halmos 1982, pp. 5−7, 168−170.

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.

[FOOTNOTEHalmos19825−7-1] Halmos 1982, pp. 5−7.

[FOOTNOTEHalmos19825−7,_168−170-2] Halmos 1982, pp. 5−7, 168−170.

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[2]

v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector–valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold