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.

Integral and derivative

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

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 $L^{p}$ 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

References

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