Functions, Functionals, and Operators

Function: $x \mapsto f(x)$

• Example: $f(x) = x^2$

Functional: $f \mapsto J[f]$

• Example: $J[f] = \int_a^b f(x)\,dx$

Operator: $f \mapsto (\mathcal{L}f)(x)$

• Example: $(\mathcal{L}f)(x) = f'(x)$

Function Spaces

Function Space: $M = \{\, f : D \to \mathbb{R} \mid \text{condition} \,\}$

• Example: $C^p[a,b] = \left\{\, f : [a,b] \to \mathbb{R} \;\middle|\; f^{(k)} \in C[a,b] \ \ \forall\, k \in \{0,1,\dots,p\} \,\right\}$
• Example: $L^p[a,b] = \left\{\, f : [a,b] \to \mathbb{R} \;\middle|\; \|f\|_{L^p[a,b]} < \infty \,\right\}$

Function Norms

Function Norm: $\|f\| : M \to \mathbb{R}_{\geq 0}$

• Example: $\|f\|{C^p[a,b]} = \sum{k=0}^{p} \max_{x \in [a,b]} \left| f^{(k)}(x) \right|$
• Example: $\|f\|_{L^p[a,b]} = \left( \int_a^b |f(x)|^p \, dx \right)^{1/p}$

Inner Products

Inner Product: $\langle f, g \rangle = \int_a^b f(x)\,g(x)\,dx \approx \sum_{i=1}^{n} f_i\, g_i\, \Delta x = (\mathbf{f} \cdot \mathbf{g})\, \Delta x$

Basis Expansions

Basis Functions: $f(x) = \sum_{i=0}^{\infty} c_i\, \phi_i(x) \approx \sum_{i=0}^{n} c_i\, \phi_i(x)$

• Example: Polynomial basis: $\phi_i(x) \in \{1, x, x^2, \dots, x^n\}$