Infinite-dimensional vector function

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

$$f(t)\; =\; (f\_1(t),\; f\_2(t),\; f\_3(t),\; \backslash ldots)\backslash ,\; ,$$

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

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

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 $A\; \backslash to\; K$ 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.

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.

If $f\; :\; [0,1]\; \backslash to\; X,$ where $X$ is a Banach space or another topological vector space then the derivative of $f$ can be defined in the usual way:

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

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:

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

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, $t\; \backslash in\; R^n$ or even $t\backslash in\; Y,$ 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

$$f\; =\; (f\_1,f\_2,f\_3,\backslash ldots)$$

(that is, $f\; =\; f\_1\; e\_1+f\_2\; e\_2+f\_3\; e\_3+\backslash cdots,$ where $e\_1,e\_2,e\_3,\backslash ldots$ is an orthonormal basis of the space $X$), and $f\text{'}(t)$ exists, then

$$f\text{'}(t)\; =\; (f\_1\text{'}(t),f\_2\text{'}(t),f\_3\text{'}(t),\backslash ldots).$$

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.

{{Main|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 $[a,\; b]$.{{sfn|Halmos|1982|pp=5−7}}

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.{{sfn|Halmos|1982|pp=5−7}}

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.{{sfn|Halmos|1982|pp=5−7}}

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:{{sfn|Halmos|1982|pp=5−7,168−170}}

$$\backslash begin\{alignat\}\{4\}$$

f :\;&& [0, 1] &&\;\to \;& L^2(0, 1) \\[0.3ex]

&& t &&\;\mapsto\;& \mathbb{1}_{[0,t]} \\

\end{alignat}

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

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 $L^2(0,\; 1).${{sfn|Halmos|1982|pp=5−7,168−170}}

A crinkled arc $f\; :\; [0,\; 1]\; \backslash to\; X$ is said to be normalized if $f(0)\; =\; 0,$ $\backslash |f(1)\backslash |\; =\; 1,$ and the span of its image $f([0,\; 1])$ is a dense subset of $X.${{sfn|Halmos|1982|pp=5−7,168−170}}

{{Math theorem|name=Proposition{{sfn|Halmos|1982|pp=5−7,168−170}}|math_statement=

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

}}

If $h\; :\; [0,\; 1]\; \backslash to\; [0,\; 1]$ is an increasing homeomorphism then $f\; \backslash circ\; h$ is called a reparameterization of the curve $f\; :\; [0,\; 1]\; \backslash to\; X.${{sfn|Halmos|1982|pp=5−7}}

Two curves $f$ and $g$ in an inner product space $X$ are unitarily equivalent if there exists a unitary operator $L\; :\; X\; \backslash to\; X$ (which is an isometric linear bijection) such that $g\; =\; L\; \backslash circ\; f$ (or equivalently, $f\; =\; L^\{-1\}\; \backslash circ\; g$).

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

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.

• {{annotated link|Differentiation in Fréchet spaces}}
• {{annotated link|Differentiable vector–valued functions from Euclidean space}}

{{reflist}}

{{reflist|group=note}}

• Einar Hille & Ralph Phillips: "Functional Analysis and Semi Groups", Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R.I., 1957.
• {{Halmos A Hilbert Space Problem Book 1982}}

{{Analysis in topological vector spaces}}

{{Functional analysis}}

{{DEFAULTSORT:Infinite-dimensional vector function}}

Category:Banach spaces

Category:Differential calculus

Category:Hilbert spaces

Category:Topological vector spaces

Category:Vectors (mathematics and physics)