8 Functional Analysis
Functional analysis is the study of function spaces and the operators acting on function spaces. In functional analysis, functions are viewed as points in a space. The main applications, in the context of physics, are the solution of differential and integral equations.
8.1 Elementary Example
8.1.1 Simple
Functional analysis treats functions as points of a space. Here three functions form a finite set of points.
\[ X = \{ f,\ g,\ h \} \]
\[ f(t) = 0,\quad g(t) = 1,\quad h(t) = t \]
where
- \(X\) is a set of functions viewed as points.
- \(t\) is the variable of each function.
8.1.2 General
Operators act on those points. A linear operator \(T\) can map each sample function to another function.
\[ T : X \rightarrow Y \]
\[ T(f) = \dfrac{df}{dt},\quad \text{on differentiable functions} \]
where
- \(T\) is an operator on a function space.
- \(\dfrac{df}{dt}\) is the derivative of \(f\).