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Banach space
A Banach space $(X,\lVert\,\cdot\,\rVert)$ is a normed vector space such that $X$ is complete under the metric induced by the norm $\lVert\,\cdot\,\rVert$.
Some authors use the term Banach space only in the case where $X$ is infinitedimensional, although on Planetmath finitedimensional spaces are also considered to be Banach spaces.
If $Y$ is a Banach space and $X$ is any normed vector space, then the set of continuous linear maps $f\colon X\to Y$ forms a Banach space, with norm given by the operator norm. In particular, since $\mathbb{R}$ and $\mathbb{C}$ are complete, the continuous linear functionals on a normed vector space $\mathcal{B}$ form a Banach space, known as the dual space of $\mathcal{B}$.
Examples:

$L^{p}$ spaces are by far the most common example of Banach spaces.

$\ell^{p}$ spaces are $L^{p}$ spaces for the counting measure on $\mathbb{N}$.

Continuous functions on a compact set under the supremum norm.
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examples by matte ✓
dual space by matte ✓
first example by matte ✓
Metric? by jacou ✘