Infinite-dimensional vector function

An infinite-dimensional vector function is a

.

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

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

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

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

.

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:

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:

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

(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 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

need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.

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

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 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] where ${\displaystyle \mathbb {1} _{[0,\,t]}:(0,1)\to \{0,1\}}$ is the indicator function defined by A crinkled arc can be found in every infinite−dimensional Hilbert space because any such space contains a 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 ${\displaystyle f([0,1])}$ is a dense subset of ${\displaystyle X.}$^[2]

If ${\displaystyle h:[0,1]\to [0,1]}$ is an

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

.

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

. Also ${\displaystyle L^{p}}$ 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.
• OCLC 8169781
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