Banach space

English

Etymology

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Noun

1. Template:Lb A normed vector space which is complete with respect to the norm, meaning that Cauchy sequences have well-defined limits that are points in the space.
   • 1962 [Prentice-Hall], Kenneth Hoffman, Banach Spaces of Analytic Functions, 2007, Dover, page 138,

     Before taking up the extreme points for $H^1$ and $H^{\mathrm{\infty }}$, let us make a few elementary observations about the unit ball $\mathrm{\Sigma }$ in the Banach space $X$.

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   • 2013, R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Template:W, page 35,

     [A] Banach space is a complete normed linear space $X$. Its dual space $X^{\prime }$ is the linear space of all continuous linear functionals $f:X\to ℝ$, and it has norm ${\Vert f\Vert }_{X^{\prime }}\equiv \text{sup}\{|f(x)|:\Vert x\Vert \le 1\}$; $X^{\prime }$ is also a Banach space.

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