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.

Set ${\displaystyle f_{k}(t)=t/k^{2}}${\displaystyle f_{k}(t)=t/k^{2}} for every positive integer ${\displaystyle k}${\displaystyle k} and every real number ${\displaystyle t.}${\displaystyle t.} Then the function ${\displaystyle f}${\displaystyle f} defined by the formula $${\displaystyle f(t)=(f_{1}(t),f_{2}(t),f_{3}(t),\ldots ),,円}$${\displaystyle f(t)=(f_{1}(t),f_{2}(t),f_{3}(t),\ldots ),,円} takes values that lie in the infinite-dimensional vector space ${\displaystyle X}${\displaystyle X} (or ${\displaystyle \mathbb {R} ^{\mathbb {N} }}${\displaystyle \mathbb {R} ^{\mathbb {N} }}) of real-valued sequences. For example, $${\displaystyle f(2)=\left(2,{\frac {2}{4}},{\frac {2}{9}},{\frac {2}{16}},{\frac {2}{25}},\ldots \right).}$${\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 ${\displaystyle X,}${\displaystyle X,} to talk about the derivative of ${\displaystyle f,}${\displaystyle f,} it is first necessary to specify a topology on ${\displaystyle X}${\displaystyle X} or the concept of a limit in ${\displaystyle X.}${\displaystyle X.}

Moreover, for any set ${\displaystyle A,}${\displaystyle A,} there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of ${\displaystyle A}${\displaystyle A} (for example, the space of functions ${\displaystyle A\to K}${\displaystyle A\to K} with finitely-many nonzero elements, where ${\displaystyle K}${\displaystyle K} is the desired field of scalars). Furthermore, the argument ${\displaystyle t}${\displaystyle 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, ${\displaystyle X}${\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.

If ${\displaystyle f:[0,1]\to X,}${\displaystyle f:[0,1]\to X,} where ${\displaystyle X}${\displaystyle X} is a Banach space or another topological vector space then the derivative of ${\displaystyle f}${\displaystyle f} can be defined in the usual way: $${\displaystyle f'(t)=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}.}$${\displaystyle f'(t)=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}.}

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

If ${\displaystyle X}${\displaystyle X} is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if $${\displaystyle f=(f_{1},f_{2},f_{3},\ldots )}$${\displaystyle f=(f_{1},f_{2},f_{3},\ldots )} (that is, ${\displaystyle f=f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3}+\cdots ,}${\displaystyle f=f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3}+\cdots ,} where ${\displaystyle e_{1},e_{2},e_{3},\ldots }${\displaystyle e_{1},e_{2},e_{3},\ldots } is an orthonormal basis of the space ${\displaystyle X}${\displaystyle X}), and ${\displaystyle f'(t)}${\displaystyle f'(t)} exists, then $${\displaystyle f'(t)=(f_{1}'(t),f_{2}'(t),f_{3}'(t),\ldots ).}$${\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 ${\displaystyle X}${\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.

If ${\displaystyle [a,b]}${\displaystyle [a,b]} is an interval contained in the domain of a curve ${\displaystyle f}${\displaystyle f} that is valued in a topological vector space then the vector ${\displaystyle f(b)-f(a)}${\displaystyle f(b)-f(a)} is called the chord of ${\displaystyle f}${\displaystyle f} determined by ${\displaystyle [a,b]}${\displaystyle [a,b]}.^[1] If ${\displaystyle [c,d]}${\displaystyle [c,d]} is another interval in its domain then the two chords are said to be non−overlapping chords if ${\displaystyle [a,b]}${\displaystyle [a,b]} and ${\displaystyle [c,d]}${\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 ${\displaystyle L^{2}}${\displaystyle L^{2}} space ${\displaystyle L^{2}(0,1)}${\displaystyle L^{2}(0,1)} is:^[2] $${\displaystyle {\begin{alignedat}{4}f:\;&&[0,1]&&\;\to \;&L^{2}(0,1)\\[0.3ex]&&t&&\;\mapsto \;&\mathbb {1} _{[0,t]}\\\end{alignedat}}}$${\displaystyle {\begin{alignedat}{4}f:\;&&[0,1]&&\;\to \;&L^{2}(0,1)\\[0.3ex]&&t&&\;\mapsto \;&\mathbb {1} _{[0,t]}\\\end{alignedat}}} where ${\displaystyle \mathbb {1} _{[0,,円t]}:(0,1)\to \{0,1\}}${\displaystyle \mathbb {1} _{[0,,円t]}:(0,1)\to \{0,1\}} is the indicator function defined by $${\displaystyle x\;\mapsto \;{\begin{cases}1&{\text{ if }}x\in [0,t]\0円&{\text{ otherwise }}\end{cases}}}$${\displaystyle x\;\mapsto \;{\begin{cases}1&{\text{ if }}x\in [0,t]\0円&{\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 ${\displaystyle L^{2}(0,1).}${\displaystyle L^{2}(0,1).}^[2] A crinkled arc ${\displaystyle f:[0,1]\to X}${\displaystyle f:[0,1]\to X} is said to be normalized if ${\displaystyle f(0)=0,}${\displaystyle f(0)=0,} ${\displaystyle \|f(1)\|=1,}${\displaystyle \|f(1)\|=1,} and the span of its image ${\displaystyle f([0,1])}${\displaystyle f([0,1])} is a dense subset of ${\displaystyle X.}${\displaystyle X.}^[2]

If ${\displaystyle h:[0,1]\to [0,1]}${\displaystyle h:[0,1]\to [0,1]} is an increasing homeomorphism then ${\displaystyle f\circ h}${\displaystyle f\circ h} is called a reparameterization of the curve ${\displaystyle f:[0,1]\to X.}${\displaystyle f:[0,1]\to X.}^[1] Two curves ${\displaystyle f}${\displaystyle f} and ${\displaystyle g}${\displaystyle g} in an inner product space ${\displaystyle X}${\displaystyle X} are unitarily equivalent if there exists a unitary operator ${\displaystyle L:X\to X}${\displaystyle L:X\to X} (which is an isometric linear bijection) such that ${\displaystyle g=L\circ f}${\displaystyle g=L\circ f} (or equivalently, ${\displaystyle f=L^{-1}\circ g}${\displaystyle f=L^{-1}\circ g}).

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

The most important integrals of ${\displaystyle f}${\displaystyle f} are called Bochner integral (when ${\displaystyle X}${\displaystyle X} is a Banach space) and Pettis integral (when ${\displaystyle X}${\displaystyle X} is a topological vector space). Both these integrals commute with linear functionals. Also ${\displaystyle L^{p}}${\displaystyle L^{p}} spaces have been defined for such functions.

• Differentiation in Fréchet spaces
• Differentiable vector–valued functions from Euclidean space – Differentiable function in functional analysisPages displaying short descriptions of redirect targets

• 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.

• Banach spaces
• Differential calculus
• Hilbert spaces
• Topological vector spaces
• Vectors (mathematics and physics)

• Pages displaying short descriptions of redirect targets via Module:Annotated link