sage: VectorSpace(QQ, 5, inner_product_matrix=matrix.identity(5)).category()
Category of finite dimensional vector spaces with basis over (number fields and quotient fields and metric spaces)

As noted in #30061, it is not in the category MetricSpaces, and thus the element methods dist and abs (?!) are missing...

The methods __abs__, norm, and dot_product are unrelated to the inner product matrix; only inner_product uses inner_product_matrix.

We define a category NormedVectorSpaces with element method abs.

We deprecate the MetricSpaces element method abs.

Follow-up: #30218: Category InnerProductSpaces

Random references:

• norms on tensor products:
  • https://www.ams.org/journals/mcom/1972-26-118/S0025-5718-1972-0305099-X/S0025-5718-1972-0305099-X.pdf
  • http://inmabb.criba.edu.ar/revuma/pdf/v52n2/v52n2a04.pdf

Depends on #30062

CC: @tscrim @simonbrandhorst @egourgoulhon @mjungmath

Component: linear algebra

Branch/Commit: public/categories/normed_monoids-30092 @ 656612e

Issue created by migration from https://trac.sagemath.org/ticket/30092