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 ${\displaystyle f_{k}(t)=t/k^{2}}$ for every positive integer ${\displaystyle k}$ and every real number ${\displaystyle t.}$ Then the function ${\displaystyle f}$ defined by the formula

$${\displaystyle f(t)=(f_{1}(t),f_{2}(t),f_{3}(t),\ldots )\,,}$$

takes values that lie in the infinite-dimensional vector space ${\displaystyle X}$ (or ${\displaystyle \mathbb {R} ^{\mathbb {N} }}$) of real-valued sequences. For example,

$${\displaystyle f(2)=\left(2,{\frac {2}{4}},{\frac {2}{9}},{\frac {2}{16}},{\frac {2}{25}},\ldots \right).}$$

As a number of different topologies can be defined on the space ${\displaystyle X,}$ 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.

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.

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:

$${\displaystyle f'(t)=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}.}$$

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:

$${\displaystyle f'(t)=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}.}$$

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, ${\displaystyle t\in R^{n}}$ 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

$${\displaystyle f=(f_{1},f_{2},f_{3},\ldots )}$$

(that is, ${\displaystyle f=f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3}+\cdots ,}$ where ${\displaystyle e_{1},e_{2},e_{3},\ldots }$ is an orthonormal basis of the space ${\displaystyle X}$), and ${\displaystyle f'(t)}$ exists, then

$${\displaystyle f'(t)=(f_{1}'(t),f_{2}'(t),f_{3}'(t),\ldots ).}$$

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 ${\displaystyle X}$ 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.

Crinkled arcs

Main article: Crinkled arc

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]

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

where ${\displaystyle \mathbb {1} _{[0,\,t]}:(0,1)\to \{0,1\}}$ is the indicator function defined by

$${\displaystyle x\;\mapsto \;{\begin{cases}1&{\text{ if }}x\in [0,t]\\0&{\text{ otherwise }}\end{cases}}}$$

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 ${\displaystyle L^{2}(0,1).}$ ^[2] A crinkled arc ${\displaystyle f:[0,1]\to X}$ is said to be normalized if ${\displaystyle f(0)=0,}$${\displaystyle \|f(1)\|=1,}$ and the span of its image ${\displaystyle f([0,1])}$ is a dense subset of ${\displaystyle X.}$ ^[2]

Proposition ^[2]  — Given any two normalized crinkled arcs in a Hilbert space, each is unitarily equivalent to a reparameterization of the other.

If ${\displaystyle h:[0,1]\to [0,1]}$ 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, ${\displaystyle f=L^{-1}\circ g}$).

Measurability

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

Integrals

The most important integrals of ${\displaystyle f}$ 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.

See also

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

References

1. Halmos 1982, pp. 5−7.
2. 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.