Hilbert space

A Hilbert space is an inner product space $(H,\langle\cdot,\cdot\rangle)$ that is complete with respect to the induced norm $\|x\|=\sqrt{\langle x,x\rangle}$.

With this induced norm, every Hilbert space is a Banach space . The Cauchy–Schwarz inequality is a fundamental tool for relating inner products to norms in Hilbert space geometry.

Examples:

• $\mathbb{R}^n$ with the usual dot product.
• For a measure space $(X,\Sigma,\mu)$, the space $L^2(X,\mu)$ of square-integrable functions with inner product $\langle f,g\rangle=\int_X f\,\overline{g}\,d\mu$.