Hilbert & Banach Space

A metric space is complete iff every Cauchy sequence in the space converges to a limit within the space.

Banach Space

A complete normed vector space.

Formally, a normed space $(X, \|\cdot\|)$ is a Banach space iff every Cauchy sequence in $X$ converges in $X$.

Examples:

• $\mathbb{R}^n$ with the Euclidean norm
• $\ell^p$ spaces for $1 \le p \le \infty$: sequences $(x_1, x_2, \ldots)$ with $\sum |x_i|^p < \infty$
• $L^p([a,b])$: functions on $[a,b]$ with finite $p$-norm

Hilbert Space

A complete inner product space. Every Hilbert space is a Banach space (the inner product induces a norm), but not every Banach space is a Hilbert space.

A norm $\|\cdot\|$ comes from an inner product iff it satisfies the parallelogram law:

Examples:

• $\mathbb{R}^n$ with the dot product
• $L^2([a,b])$: square-integrable functions with inner product $\langle f, g \rangle = \int_a^b f(x)\overline{g(x)}\,\text{d}x$
• $\ell^2$: square-summable sequences with inner product $\langle x, y \rangle = \sum x_i \overline{y_i}$

Relation