Hilbert spaces

An inner product space $$V$$ that is a Banach space equipped with the norm (Def 4.1.5)

$$||\cdot || = \sqrt{\langle \cdot, \cdot \rangle}$$

is called a Hilbert space. A Hilbert space is typically denoted as $$\mathcal{H}$$. Clearly, by definition Hilbert spaces are a subset of Banach spaces.

Examples to Hilbert spaces are $$\mathbb{R}^n, \mathbb C^n$$ and $$\ell^2(\mathbb{N})$$ (see Banach spaces for the latter).