Notes on Normed Linear & Banach Space
Normed space and Banach space.
A norm is a nonnegative homogeneous mapping on a vector space that satisfies the triangle axiom: for a vector space $X$ with underlying field $\mathbb{F}$, a mapping $||: X \to \mathbb{R}_{\ge 0}$
A normed space is a vector space with a norm. Norm specifies the length of each element of a vector space. A norm induces a metric on the vector space, $d(x,y) = |x − y|$.
A Banach space is a complete normed space (in the induced metric).
Sobolev space $W^{s,p}(\Omega)$ with norm $|f|_{s,p,\Omega} = \sum_{|\alpha| \le s} |\partial_x^\alpha f|_{L^p(\Omega)}$ is a Banach space.