12 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.
12.2 Examples
12.2.1 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.
12.2.2 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 \(f\).
- Absolute-Value-to-the-p
- Banach Space
- Cauchy Sequence
- Complete Set
- Completeness
- Convergence
- Delta Function
- Ell P
- Fourier Series
- Fourier Transform
- Fourier-Stieltjes Coefficients
- Functional
- Integrability
- Linear Space
- Metric Space
- Norm
- Normed Space
- P
- P-Norm
- P-Summable
- P-Summable Sequence
- Quadratic Integrability
- Sets
- Sifting Property