# 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 ${\displaystyle f_{k}(t)=t/k^{2}}$ for every positive integer k and every real number t. Then values of the function

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

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

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

As a number of different topologies can be defined on the space X, we cannot talk about the derivative of f without first defining the topology of 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 (e.g., the space of functions ${\displaystyle A\rightarrow K}$ 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, e.g., ${\displaystyle f:[0,1]\rightarrow X}$, where X is a Banach space or another topological vector space, the derivative of f can be defined in the standard way: ${\displaystyle f'(t):=\lim _{h\rightarrow 0}{\frac {f(t+h)-f(t)}{h}}}$.

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 ${\displaystyle L^{p}}$ 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, e.g., 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:

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

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., ${\displaystyle t\in R^{n}}$ or even ${\displaystyle t\in Y}$, 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

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

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

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

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.

### 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, e.g., 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.

## References

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