Infinite-dimensional vector function

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.

Such functions are applied in most sciences including physics.

Example

Set $f_{k}(t)=t/k^{2}$ for every positive integer k and every real number t. Then the function f, defined for real numbers t by the formula

$f(t)=(f_{1}(t),f_{2}(t),f_{3}(t),\ldots )\,,$

takes values that lie in the infinite-dimensional vector space X (or $\mathbb {R} ^{\mathbb {N} }$) of real-valued sequences. For example,

$f(2)=\left(2,{\frac {2}{4}},{\frac {2}{9}},{\frac {2}{16}},{\frac {2}{25}},\ldots \right).$

As a number of different topologies can be defined on the space X,to talk about the derivative of f, it is first necessary to specify a topology on X or the concept of a limit in X.

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

Integral and derivative

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

The measurability of 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 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 functionals. Also $L^{p}$ spaces have been defined for such functions.

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 is a Hilbert space); see Radon–Nikodym theorem

Derivative

Functions with values in a Hilbert space

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:

$f'(t)=\lim _{h\to 0}{\frac {f(t+h)-f(t)}{h}}.$

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, $t\in R^{n}$ or even $t\in Y,$ where Y is an infinite-dimensional vector space).

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

$f=(f_{1},f_{2},f_{3},\ldots )$

(that is, $f=f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3}+\cdots ,$ where $e_{1},e_{2},e_{3},\ldots$ is an orthonormal basis of the space X), and $f'(t)$ exists, then

$f'(t)=(f_{1}'(t),f_{2}'(t),f_{3}'(t),\ldots ).$

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.

Other infinite-dimensional vector spaces

Most of the above hold for other topological vector spaces 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.

See also

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

References

• Einar Hille & Ralph Phillips: "Functional Analysis and Semi Groups", Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R.I., 1957.