Categories of normed additive monoids/groups, normed modules/rings/fields, add FreeQuadraticModule to normed vector spaces

[mkoeppe]

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

[mkoeppe]

comment:1

Also, the classes in sage.modules do not make it possible to pass a category argument at initialization.

[mkoeppe]

Description changed:

--- 
+++ 
@@ -4,3 +4,8 @@
 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`.
+
+

[mkoeppe]

Dependencies: #30062

[mkoeppe]

Description changed:

--- 
+++ 
@@ -9,3 +9,10 @@
 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 `InnerProductSpaces` with element methods `inner_product`.
+
+We define a category `NormedVectorSpaces` with element method `abs`.
+
+We deprecate the `MetricSpaces` element method `abs`.
+
+

[mkoeppe]

Branch: u/mkoeppe/category_inner_product_spaces

[mkoeppe]

Commit: 26e9c8f

[mkoeppe]

comment:6

I have here a preliminary implementation (trying to learn from FilteredModules and MetricSpaces) of NormedSpacesCategory and NormedVectorSpaces.

Travis, if you could take a quick look to see if this seems to be going in the right direction...

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New commits:

1f4f942	Provide EuclideanSpace with dist() method and add it to category of metric space
7a92b60	sage.categories.metric_spaces.MetricSpaces: Rename metric to metric_function with deprecation
3ee003a	Merge branch 't/30062/public/manifolds/euclidean_metric_space' into t/30092/category_inner_product_spaces
26e9c8f	NormedVectorSpaces: New

[tscrim]

comment:7

Why is this a functorial construction and not an axiom? I don't see why this wouldn't commute with other axioms, such as WithBasis.

[mkoeppe]

comment:8

Thanks. I'll look into this.

[tscrim]

comment:9

Ah, no, this is good. I thought I implemented metric spaces as an axiom, but it is a functorial construction. So please keep the current approach.

[mkoeppe]

comment:10

Thanks. I am still quite confused about many basic things in the category system, thanks for your patience. I am getting this:

sage: VectorSpaces(QQ).Filtered().Normed()
Join of Category of filtered vector spaces over Rational Field and Category of metric spaces

What am I missing for "normed spaces" to show up?

[tscrim]

comment:11

Try

-class NormedSpacesCategory(RegressiveCovariantConstructionCategory, Category_over_base_ring):
+class NormedSpacesCategory(RegressiveCovariantConstructionCategory):

[tscrim]

comment:12

Also remove the __init__.

[sagetrac-git]

Changed commit from 26e9c8f to 5eee4c1

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

8651b8b	Fix syntax
5eee4c1	Add doc