Aninfinite-dimensional vector function is afunction whose values lie in aninfinite-dimensionaltopological vector space, such as aHilbert space or aBanach space.

Such functions are applied in most sciences includingphysics.

Set${\displaystyle f_k(t)=t/k^2}$ for every positiveinteger${\displaystyle k}$ and everyreal number${\displaystyle t.}$ Then the function${\displaystyle f}$ defined by the formula$${\displaystyle f(t)=(f_1(t),f_2(t),f_3(t),\dots ),}$$takes values that lie in the infinite-dimensionalvector space${\displaystyle X}$ (or${\displaystyle {\mathbb{R}}^{\mathbb{N}}}$) of real-valuedsequences. For example,$${\displaystyle f(2)=\left(2,\frac{2}{4},\frac{2}{9},\frac{2}{16},\frac{2}{25},\dots \right).}$$

As a number of differenttopologies can be defined on the space${\displaystyle X,}$ to talk about thederivative of${\displaystyle f,}$ it is first necessary to specify a topology on${\displaystyle X}$ or the concept of alimit in${\displaystyle X.}$

Moreover, for any set${\displaystyle A,}$ there exist infinite-dimensional vector spaces having the (Hamel)dimension of thecardinality of${\displaystyle A}$ (for example, the space of functions${\displaystyle A\to K}$ with finitely-many nonzero elements, where${\displaystyle K}$ is the desiredfield of scalars). Furthermore, the argument${\displaystyle t}$ could lie in any set instead of the set of real numbers.

Most theorems onintegration anddifferentiation of scalar functions can be generalized to vector-valued functions, often using essentially the sameproofs. Perhaps the most important exception is thatabsolutely continuous functions need not equal the integrals of their (a.e.) derivatives (unless, for example,${\displaystyle X}$ is a Hilbert space); seeRadon–Nikodym theorem

Acurve is acontinuous map of theunit interval (or more generally, of a non−degenerateclosedinterval of real numbers) into atopological space. Anarc is a curve that is also atopological embedding. A curve valued in aHausdorff space is an arcif and only if it isinjective.

If${\displaystyle f:[0,1]\to X,}$ where${\displaystyle X}$ is a Banach space or anothertopological vector space then thederivative of${\displaystyle f}$ can be defined in the usual way:$${\displaystyle f^{\prime }(t)=\underset{h\to 0}{lim}\frac{f(t+h)-f(t)}{h}.}$$

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:$${\displaystyle f^{\prime }(t)=\underset{h\to 0}{lim}\frac{f(t+h)-f(t)}{h}.}$$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,${\displaystyle t\in R^n}$ 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$${\displaystyle f=(f_1,f_2,f_3,\dots )}$$(that is,${\displaystyle f=f_1{e}_1+f_2{e}_2+f_3{e}_3+\cdots ,}$ where${\displaystyle e_1,e_2,e_3,\dots }$ is anorthonormal basis of the space${\displaystyle X}$), and${\displaystyle f^{\prime }(t)}$ exists, then$${\displaystyle f^{\prime }(t)=(f_1^{\prime }(t),f_2^{\prime }(t),f_3^{\prime }(t),\dots ).}$$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 othertopological vector spaces${\displaystyle X}$ too. However, not as many classical results hold in theBanach space setting, for example, anabsolutely continuous function with values in asuitable Banach space need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.

If${\displaystyle [a,b]}$ is an interval contained in thedomain of a curve${\displaystyle f}$ that is valued in atopological vector space then the vector${\displaystyle f(b)-f(a)}$ is called thechord 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 benon−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 aninner product space areorthogonal vectors if the curve makes aright 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 bedifferentiable at any point.^[1] Acrinkled 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 theHilbert${\displaystyle L^2}$ space${\displaystyle L^2(0,1)}$ is:^[2]where${\displaystyle {\mathbb{1}}_{[0,t]}:(0,1)\to \{0,1\}}$ is theindicator function defined byA crinkled arc can be found in every infinite−dimensional Hilbert space because any such space contains aclosedvector subspace that isisomorphic to${\displaystyle L^2(0,1).}$^[2] A crinkled arc${\displaystyle f:[0,1]\to X}$ is said to benormalized if${\displaystyle f(0)=0,}$${\displaystyle \Vert f(1)\Vert =1,}$ and thespan of itsimage${\displaystyle f([0,1])}$ is adense subset of${\displaystyle X.}$^[2]

If${\displaystyle h:[0,1]\to [0,1]}$ is anincreasinghomeomorphism then${\displaystyle f\circ h}$ is called areparameterization of the curve${\displaystyle f:[0,1]\to X.}$^[1] Two curves${\displaystyle f}$ and${\displaystyle g}$ in aninner product space${\displaystyle X}$ areunitarily equivalent if there exists aunitary operator${\displaystyle L:X\to X}$ (which is anisometriclinearbijection) such that${\displaystyle g=L\circ f}$ (or equivalently,${\displaystyle f=L^{-1}\circ g}$).

Themeasurability of${\displaystyle f}$ can be defined by a number of ways, most important of which areBochner measurability andweak measurability.

The most important integrals of${\displaystyle f}$ are calledBochner integral (when${\displaystyle X}$ is a Banach space) andPettis integral (when${\displaystyle X}$ is a topological vector space). Both these integrals commute withlinear functionals. Also${\displaystyle L^p}$ spaces have beendefined 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)

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