L2-Space

A Hilbert Space in which a Bracket Product is defined by

$$\left\langle{\phi\vert\psi}\right\rangle{} \equiv \int \psi^*\phi \,dx$$ (1)

and which satisfies the following conditions

$$\left\langle{\phi\vert\psi}\right\rangle{}^* = \left\langle{\psi\vert\phi}\right\rangle{}e$$ (2)

$$\left\langle{\phi\vert\lambda_1\psi_1+\lambda_2\psi_2}\right... ...rangle{}+\lambda_2\left\langle{\phi\vert\psi_2}\right\rangle{}$$ (3)

$$\left\langle{\lambda_1\phi_1+\lambda_2\phi_2\vert\psi}\right... ...le{}+{\lambda_2}^*\left\langle{\phi_2\vert\psi}\right\rangle{}$$ (4)

$$\left\langle{\psi\vert\psi}\right\rangle{} \in \Bbb{R}\geq 0$$ (5)

$$\vert\left\langle{\psi_1\vert\psi_2}\right\rangle{}\vert^2 \... ...\right\rangle{}\left\langle{\psi_2\vert\psi_2}\right\rangle{}.$$ (6)

The last of these is Schwarz's Inequality.

See also Bracket Product, Hilbert Space, L2-Norm, Riesz-Fischer Theorem, Schwarz's Inequality