\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Needs:
Complete Metric Spaces ›
Norm Metrics ›
Needed by:
Complete Real Inner Product Spaces ›
Dual Spaces ›
Links:
Sheet PDF ›
Graph PDF ›

Complete Normed Spaces

Definition

A complete normed space (or Banach space, or complete norm space) is a normed space whose induced metric space is complete.1

  1. The term Banach space is universal, but in accordance with the Bourbaki project’s guidelines on naming, we will tend to use the term complete normed space, even though such terminology is more verbose. ↩︎
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