View source for Infinite-dimensional vector function

{{Short description|Whose values lie in an infinite-dimensional vector space}}
An '''infinite-dimensional vector function''' is a [[Function (mathematics)|function]] whose values lie in an infinite-dimensional [[Topological vector space|topological vector space]], such as a [[Hilbert space]] or a [[Banach space]].

Such functions are applied in most sciences including [[HandWiki:Physics|physics]].

==Example==

Set <math>f_k(t) = t/k^2</math> for every positive [[Integer|integer]] <math>k</math> and every [[Real number|real number]] <math>t.</math> Then the function <math>f</math> defined by the formula
<math display=block>f(t) = (f_1(t), f_2(t), f_3(t), \ldots)\, ,</math>
takes values that lie in the infinite-dimensional [[Vector space|vector space]] <math>X</math> (or <math>\R^{\N}</math>) of real-valued [[Sequence|sequence]]s. For example,
<math display=block>f(2) = \left(2, \frac{2}{4}, \frac{2}{9}, \frac{2}{16}, \frac{2}{25}, \ldots\right).</math>

As a number of different [[Topological space|topologies]] can be defined on the space <math>X,</math> to talk about the [[Derivative|derivative]] of <math>f,</math> it is first necessary to specify a topology on <math>X</math> or the concept of a [[Limit (mathematics)|limit]] in <math>X.</math>

Moreover, for any set <math>A,</math> there exist infinite-dimensional vector spaces having the (Hamel) [[Dimension (vector space)|dimension]] of the [[Cardinality|cardinality]] of <math>A</math> (for example, the space of functions <math>A \to K</math> with finitely-many nonzero elements, where <math>K</math> is the desired [[Field (mathematics)|field]] of scalars). Furthermore, the argument <math>t</math> could lie in any set instead of the set of real numbers.

==Integral and derivative==

Most theorems on [[Integral|integration]] and [[Derivative|differentiation]] of scalar functions can be generalized to vector-valued functions, often using essentially the same [[Mathematical proof|proofs]]. Perhaps the most important exception is that absolutely continuous functions need not equal the integrals of their (a.e.) derivatives (unless, for example, <math>X</math> is a Hilbert space); see [[Radon–Nikodym theorem]]

A curve is a continuous map of the [[Unit interval|unit interval]] (or more generally, of a non−degenerate closed [[Interval (mathematics)|interval]] of real numbers) into a [[Topological space|topological space]]. An ''[[Arc (topology)|arc]]'' is a curve that is also a topological embedding. A curve valued in a [[Hausdorff space]] is an arc [[If and only if|if and only if]] it is [[Injective function|injective]]. 

===Derivatives===

If <math>f : [0,1] \to X,</math> where <math>X</math> is a Banach space or another [[Topological vector space|topological vector space]] then the [[Derivative|derivative]] of <math>f</math> can be defined in the usual way: 
<math display=block>f'(t) = \lim_{h\to 0}\frac{f(t+h)-f(t)}{h}.</math>

====Functions with values in a Hilbert space====

If <math>f</math> is a function of real numbers with values in a Hilbert space <math>X,</math> then the derivative of <math>f</math> at a point <math>t</math> can be defined as in the finite-dimensional case:
<math display=block>f'(t)=\lim_{h\to 0} \frac{f(t+h)-f(t)}{h}.</math>
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, <math>t \in R^n</math> or even <math>t\in Y,</math> where <math>Y</math> is an infinite-dimensional vector space).

If <math>X</math> is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if
<math display=block>f = (f_1,f_2,f_3,\ldots)</math>
(that is, <math>f = f_1 e_1+f_2 e_2+f_3 e_3+\cdots,</math> where <math>e_1,e_2,e_3,\ldots</math> is an [[Orthonormal basis|orthonormal basis]] of the space <math>X</math>), and <math>f'(t)</math> exists, then
<math display=block>f'(t) = (f_1'(t),f_2'(t),f_3'(t),\ldots).</math>
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 space|topological vector space]]s <math>X</math> 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|Crinkled arc}}

If <math>[a, b]</math> is an interval contained in the [[Domain of a function|domain]] of a curve <math>f</math> that is valued in a [[Topological vector space|topological vector space]] then the vector <math>f(b) - f(a)</math> is called the ''chord of <math>f</math> determined by <math>[a, b]</math>''.{{sfn|Halmos|1982|pp=5−7}} 
If <math>[c, d]</math> is another interval in its domain then the two chords are said to be ''non−overlapping chords'' if <math>[a, b]</math> and <math>[c, d]</math> 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|inner product space]] are orthogonal vectors if the curve makes a [[Right angle|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 space|Hilbert]] [[Lp space|<math>L^2</math> space]] <math>L^2(0, 1)</math> is:{{sfn|Halmos|1982|pp=5−7,168−170}}
<math display=block>\begin{alignat}{4}
f :\;&& [0, 1] &&\;\to    \;& L^2(0, 1) \\[0.3ex]
     && t      &&\;\mapsto\;& \mathbb{1}_{[0,t]} \\
\end{alignat}</math>
where <math>\mathbb{1}_{[0,\,t]} : (0, 1) \to \{0, 1\}</math> is the [[Indicator function|indicator function]] defined by
<math display=block>x \;\mapsto\; \begin{cases}1 & \text{ if } x \in [0, t]\\ 0 & \text{ otherwise } \end{cases}</math>
A crinkled arc can be found in every infinite−dimensional Hilbert space because any such space contains a [[Closed set|closed]] vector subspace that is isomorphic to <math>L^2(0, 1).</math>{{sfn|Halmos|1982|pp=5−7,168−170}} 
A crinkled arc <math>f : [0, 1] \to X</math> is said to be ''normalized'' if <math>f(0) = 0,</math> <math>\|f(1)\| = 1,</math> and the span of its [[Image (mathematics)|image]] <math>f([0, 1])</math> is a [[Dense set|dense subset]] of <math>X.</math>{{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 <math>h : [0, 1] \to [0, 1]</math> is an [[Increasing function|increasing]] [[Homeomorphism|homeomorphism]] then <math>f \circ h</math> is called a ''reparameterization'' of the curve <math>f : [0, 1] \to X.</math>{{sfn|Halmos|1982|pp=5−7}} 
Two curves <math>f</math> and <math>g</math> in an [[Inner product space|inner product space]] <math>X</math> are ''unitarily equivalent'' if there exists a [[Unitary operator|unitary operator]] <math>L : X \to X</math> (which is an [[Isometry|isometric]] [[Linear map|linear]] [[Bijection|bijection]]) such that <math>g = L \circ f</math> (or equivalently, <math>f = L^{-1} \circ g</math>).

===Measurability===

The [[Measurable function|measurability]] of <math>f</math> can be defined by a number of ways, most important of which are Bochner measurability and weak measurability.

===Integrals===

The most important integrals of <math>f</math> are called [[Bochner integral]] (when <math>X</math> is a Banach space) and [[Pettis integral]] (when <math>X</math> is a topological vector space). Both these integrals commute with [[Linear functional|linear functional]]s. Also <math>L^p</math> spaces have been [[Bochner space|defined]] for such functions.

==See also==

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

==References==

{{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}} <!--{{sfn|Halmos|1982|pp=}}-->

{{Analysis in topological vector spaces}}
{{Functional analysis}}

{{DEFAULTSORT:Infinite-dimensional vector function}}
[[Category:Banach spaces]]
[[Category:Differential calculus]]
[[Category:Topological vector spaces]]
[[Category:Vectors (mathematics and physics)]]

{{Sourceattribution|Infinite-dimensional vector function}}