As I understand, one typically derives properties of functions that satisfy certain functional equations by substituting x for various values like 0, 1, y, -x, x^2 and so on. But what I am wondering is why algebraically solving for the unknown function doesn't work, or at least what it means when you do.
Perhaps some clarification of terminology is in order.
Some authors use the term 'functional analysis' to mean the theory of functions of a (real or complex) variable.
That is the word 'functional' is used as an adjective.
I think this is the meaning intended here.
Unfortunately the word functional is also used in mathematics as a noun with a particular meaning.
This meaning was introduced by Kantorovich, Banach and Kerysig
A functional is a map from a space of test functions, , (ie functions of interest) to its underlying field.
In your case I think the underlying field is the field of Real Numbers, R
So a definite integral is such a map from the space of integrable functions to the reals and outputs a real number for each definite integral.
Functionals can be non linear or linear.
A linear real functional, F, is a map such that for any two functions,
and scalars (real numbers)
Some texts on the subject are
Kantorovich Functional Analysis
Kreysig Introductory Functional Analysis with Applications
Griffel Applied functional Analysis
Functional Analysis grew out of Dirac's version of the theory of 'generalised functions' called Distribution Theory.
Edited by studiot, 24 January 2017 - 05:55 PM.