Infinite dimensional vector function

Example

Set f k ( t ) = t / k 2 for every positive integer k and every real number t. Then values of the function

lie in the infinite-dimensional vector space X (or R N ) of real-valued sequences. For example,

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 A → 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., f : [ 0 , 1 ] → 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 → 0 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

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:

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., t ∈ R n or even t ∈ 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

(i.e., f = f 1 e 1 + f 2 e 2 + f 3 e 3 + ⋯ , where e 1 , e 2 , e 3 , … is an orthonormal basis of the space X), and f ′ ( t ) exists, then

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.