$$ \langle f, g \rangle = \int_a^b f(x) \, \bar{g}(x) \, dx \approx \sum_{i=1}^n f_i \, \bar{g}_i \left( \frac{b-a}{n-1} \right) = \langle \mathbf{f}, \mathbf{g} \rangle \Delta x $$
$$ v_i = g_i + i h_i \\ |v_i| = \sqrt{g_i^2 + h_i^2} \\ \|\mathbf{v}\|p = \left( \sum{i=1}^n |v_i|^p \right)^{\frac{1}{p}} \\ \ell^p = \left\{ \mathbf{v} \in \mathbb{C}^n \mid \|\mathbf{v}\|_p^p < \infty \right\} $$
$$ f(x) = g(x) + i h(x) \\ |f(x)| = \sqrt{g(x)^2 + h(x)^2} \\ \|f(x)\|_p = \left( \int_a^b |f(x)|^p \, dx \right)^{\frac{1}{p}} \\ \mathcal{L}^p([a,b]) = \left\{ f : [a,b] \mapsto \mathbb{C} \mid \|f(x)\|_p^p < \infty \right\} $$
$$ \mathbf{v} = \sum_{k=1}^n \frac{\langle \mathbf{v}, \mathbf{b}_k \rangle}{\|\mathbf{b}_k\|_2^2} \mathbf{b}k \\ \langle \mathbf{v}, \mathbf{b}k \rangle = \sum{i=1}^{n} v_i \, \bar{b}{k,i} \\ \|\mathbf{b}k\|2^2 = \sum{i=1}^{n} |b{k,i}|^2 $$
$$ f(x) = \int_\Omega \frac{\langle f, \psi \rangle}{\|\psi\|_2^2} \psi(\omega, x) \, d\omega \\ \hat{f}(\omega) = \langle f, \psi \rangle = \int_X f(x) \, \bar{\psi}(\omega, x) \, dx \\ \|\psi\|_2^2 = \int_X |\psi(\omega, x)|^2 \, dx $$
$$ \boxed{ \begin{array}{l l} \text{Fourier, } \hat{f}(\omega) & \psi(\omega, x) = e^{-i \omega x} \\ \text{Laplace, } \hat{f}(s) & \psi(s, t) = e^{-\sigma t} e^{-i \omega t} = e^{-st} \\ \text{Gabor, } \hat{f}(\omega, \tau) & \psi(\omega, \tau, t) = \bar{g}(t - \tau) e^{-i \omega t} \\ \text{Wavelet, } \hat{f}(a, b) & \psi(a, b, t) = \frac{1}{\sqrt{a}} \Psi\left( \frac{t - b}{a} \right) \end{array} } $$