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L2-Space

A Hilbert Space in which a Bracket Product is defined by

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

and which satisfies the following conditions
\begin{displaymath}
\left\langle{\phi\vert\psi}\right\rangle{}^* = \left\langle{\psi\vert\phi}\right\rangle{}e
\end{displaymath} (2)


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


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


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


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

The last of these is Schwarz's Inequality.

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




© 1996-9 Eric W. Weisstein
1999-05-26