Banach ring



Higher algebra



A Banach ring is a complete normed ring, hence a commutative monoid in the monoidal category of complete normed groups (with short group homomorphisms and the projective tensor product).

If not just in complete normed groups but in complete normed vector spaces (Banach spaces), then this is a Banach algebra.

The Berkovich spectrum of a Banach ring RR is the topological space of multiplicative seminorms on RR that are bounded by the norm on RR.


  • The integers \mathbb{Z} equipped with their absolute value norm || {\vert- \vert_\infty} are a Banach ring.

  • The integers with the pp-adic norm || p{|-|_p} are an incomplete normed ring whose completion is the Banach ring p\mathbb{Z}_p of pp-adic integers.

algebraic structuregroupringfieldvector spacealgebra
(submultiplicative) normnormed groupnormed ringnormed fieldnormed vector spacenormed algebra
multiplicative norm (absolute value/valuation)valued field
completenesscomplete normed groupBanach ringcomplete fieldBanach vector spaceBanach algebra


A quick review is in

  • Sarah Brodsky?, Non-archimedean geometry (pdf)

A standard textbook account in the context of rigid analytic geometry is

A set of lecture notes in the context of Berkovich spaces is

  • Vladimir Berkovich, section 1.2 of Non-archimedean analytic spaces, lectures at the Advanced School on pp-adic Analysis and Applications, ICTP, Trieste, 31 August - 11 September 2009 (pdf)

Discussion from a more topos-theoretic point of view is in

category: analysis

Last revised on July 18, 2014 at 01:22:33. See the history of this page for a list of all contributions to it.