href="#fb3_img_img_8afa513f-85fe-577d-968c-2315adec5a1c.png" alt="upper R left-parenthesis left-bracket a comma b right-bracket comma upper X right-parenthesis"/>) which does not belong to , showing that, in the infinite dimensional‐valued case, may be a proper subset of .
Example 1.44: Let be an arbitrary set and let be a normed space. A family of elements of is summable with sum (we write ), if for every , there is a finite subset such that for every finite subset with ,
Let denote the set of all families , , such that the family is summable, that is,
It is known that the expression defines an inner product and , equipped with the norm is a Hilbert space. As a consequence of the Basis Theorem, since is a Hilbert space, is a maximal orthonormal system for , that is, and stands for the Kronecker delta (see [128, Theorem 4.6], item 6, p. 61)).
In what follows, we will use the the Bessel equality given as
Let be a nondegenerate closed interval of and be equipped with the norm
Consider a function given by , . Given , there exists , with , such that for every ‐fine ,
