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

Moreover, for any set A , {\displaystyle A,} there exist infinite-dimensional vector spaces having the (Hamel) dimension of the cardinality of A {\displaystyle A} (for example, the space of functions A → K {\displaystyle A\to K} with finitely-many nonzero elements, where K {\displaystyle K} is the desired field of scalars). Furthermore, the argument 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, 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 f : [ 0 , 1 ] → X , {\displaystyle f:[0,1]\to X,} where X {\displaystyle X} is a Banach space or another topological vector space then the derivative of f {\displaystyle f} can be defined in the usual way: f ′ ( t ) = lim h → 0 f ( t + h ) − f ( t ) h . {\displaystyle f'(t)=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}.}

If f {\displaystyle f} is a function of real numbers with values in a Hilbert space X , {\displaystyle X,} then the derivative of f {\displaystyle f} at a point t {\displaystyle t} can be defined as in the finite-dimensional case: f ′ ( t ) = lim h → 0 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, t ∈ R n {\displaystyle t\in R^{n}} or even t ∈ Y , {\displaystyle t\in Y,} where Y {\displaystyle Y} is an infinite-dimensional vector space).

If X {\displaystyle X} is a Hilbert space then any derivative (and any other limit) can be computed componentwise: if f = ( f 1 , f 2 , f 3 , … ) {\displaystyle f=(f_{1},f_{2},f_{3},\ldots )} (that is, f = f 1 e 1 + f 2 e 2 + f 3 e 3 + ⋯ , {\displaystyle f=f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3}+\cdots ,} where e 1 , e 2 , e 3 , … {\displaystyle e_{1},e_{2},e_{3},\ldots } is an orthonormal basis of the space X {\displaystyle X} ), and f ′ ( t ) {\displaystyle f'(t)} exists, then f ′ ( t ) = ( f 1 ′ ( t ) , f 2 ′ ( t ) , f 3 ′ ( t ) , … ) . {\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 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 [ a , b ] {\displaystyle [a,b]} is an interval contained in the domain of a curve f {\displaystyle f} that is valued in a topological vector space then the vector f ( b ) − f ( a ) {\displaystyle f(b)-f(a)} is called the chord of f {\displaystyle f} determined by [ a , b ] {\displaystyle [a,b]} .^[1] If [ c , d ] {\displaystyle [c,d]} is another interval in its domain then the two chords are said to be non−overlapping chords if [ a , b ] {\displaystyle [a,b]} and [ 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 L 2 {\displaystyle L^{2}} space L 2 ( 0 , 1 ) {\displaystyle L^{2}(0,1)} is:^[2] f : [ 0 , 1 ] → L 2 ( 0 , 1 ) t ↦ 1 [ 0 , t ] {\displaystyle {\begin{alignedat}{4}f:\;&&[0,1]&&\;\to \;&L^{2}(0,1)\\[0.3ex]&&t&&\;\mapsto \;&\mathbb {1} _{[0,t]}\\\end{alignedat}}} where 1 [ 0 , t ] : ( 0 , 1 ) → { 0 , 1 } {\displaystyle \mathbb {1} _{[0,\,t]}:(0,1)\to \{0,1\}} is the indicator function defined by x ↦ { 1  if  x ∈ [ 0 , t ] 0  otherwise  {\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 L 2 ( 0 , 1 ) . {\displaystyle L^{2}(0,1).} ^[2] A crinkled arc f : [ 0 , 1 ] → X {\displaystyle f:[0,1]\to X} is said to be normalized if f ( 0 ) = 0 , {\displaystyle f(0)=0,} ‖ f ( 1 ) ‖ = 1 , {\displaystyle \|f(1)\|=1,} and the span of its image f ( [ 0 , 1 ] ) {\displaystyle f([0,1])} is a dense subset of X . {\displaystyle X.} ^[2]

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

The measurability of 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 f {\displaystyle f} are called Bochner integral (when X {\displaystyle X} is a Banach space) and Pettis integral (when X {\displaystyle X} is a topological vector space). Both these integrals commute with linear functionals. Also 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.