We are using the book Methods of Real Analysis by Richard R. Goldberg
(See attached file for full problem description)
Let be a complete orthogonal family in .
Define the function A from into
.( This means: In order to manufacture our metric space we must therefore regard any two function whose values are equal almost everywhere as representing the same point in our space. This is,the points in the space-which we denote by -are,
by definition, classes of square integrable functions, the functions in any one class differing from one another only on sets of measure zero.
This solution is comprised of a detailed explanation to define the function A