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| '''Infinite-dimensional vector function''' refers to a function whose values lie in an [[infinite-dimensional]] [[vector space]], such as a [[Hilbert space]] or a [[Banach space]].
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| Such functions are applied in most sciences including [[physics]].
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| ==Example==
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| Set <math>f_k(t)=t/k^2</math> for every positive integer ''k'' and every real number ''t''. Then values of the function
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| :<math>f(t)=(f_1(t),f_2(t),f_3(t),\ldots) \, </math> | |
| lie in the infinite-dimensional vector space ''X'' (or <math>\mathbf R^{\mathbf N}</math>) of real-valued sequences. For example,
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| :<math>f(2) = \left(2,\frac24,\frac29,\frac2{16},\frac2{25},\ldots\right).</math> | |
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| As a number of different topologies can be defined on the space ''X'', we cannot talk about the derivative of ''f'' without first
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| defining the topology of ''X'' or the concept of a limit in ''X''.
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| Moreover, for any set ''A'', there exist infinite-dimensional vector spaces having the (Hamel) [[vector dimension|dimension]] of the cardinality of ''A'' (e.g., the space of functions <math>A\rightarrow K</math> with finitely-many nonzero elements, where ''K'' is the desired [[Scalar field|field of scalars]]). Furthermore, the argument ''t'' could lie in any set instead of the set of real numbers.
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| ==Integral and derivative==
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| If, e.g., <math>f:[0,1]\rightarrow X</math>, where ''X'' is a Banach space or another [[topological vector space]], the derivative of ''f'' can be defined in the standard way: <math>f'(t):=\lim_{h\rightarrow0}\frac{f(t+h)-f(t)}{h}</math>.
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| The measurability of ''f'' can be defined by a number of ways, most important of which are [[Bochner measurability]] and [[weak measurability]].
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| 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 functional]]s. Also <math>L^p</math> spaces have been [[Bochner space|defined]] for such functions.
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| 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, e.g., ''X'' is a Hilbert space); see [[Radon–Nikodym theorem]]
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| ==Derivative==
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| ===Functions with values in a Hilbert space===
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| 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:
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| :<math>f'(t)=\lim_{h\rightarrow0}\frac{f(t+h)-f(t)}{h}.</math>
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| Most results of the finite-dimensional case also hold in the infinite-dimensional case too, mutatis mutandis. Differentiation can also be defined to functions of several variables (e.g., <math>t\in R^n</math> or even <math>t\in Y</math>, where ''Y'' is an infinite-dimensional vector space).
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| N.B. If ''X'' is a Hilbert space, then one can easily show that any derivative (and any other limit) can be computed componentwise: if
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| :<math>f=(f_1,f_2,f_3,\ldots)</math>
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| (i.e., <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]] of the space ''X''), and <math>f'(t)</math> exists, then
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| :<math>f'(t)=(f_1'(t),f_2'(t),f_3'(t),\ldots)</math>.
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| 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.
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| ===Other infinite-dimensional vector spaces===
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| Most of the above hold for other [[topological vector space]]s ''X'' too. However, not as many classical results hold in the [[Banach space]] setting, e.g., an [[absolutely continuous]] function with values in a [[Radon–Nikodym property|suitable Banach space]] need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.
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| ==References==
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| * Einar Hille & Ralph Phillips: "Functional Analysis and Semi Groups", Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R.I., 1957.
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| [[Category:Vectors]]
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I’m Gregorio from Porsgrunn studying Art History. I did my schooling, secured 81% and hope to find someone with same interests in Disc golf.
Look into my blog :: Here is your mountain bike sizing.