feat: a linear map which is a local embedding is an embedding

[ADedecker]

---

[github-actions]

PR summary 175663f091

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Topology.Algebra.Module.EmbeddingOfLocal (new file)	1630

---

Declarations diff (regex)

+ ContinuousSMul.topology_eq_of_induced_eq
+ ContinuousSMul.topology_eq_of_nhds_inf_principal_eq
+ Filter.IsBoundedUnder.smul_tendsto_zero_normedSpace
+ Filter.comap_inf_congr
+ Filter.comap_inf_congr_aux
+ LinearMap.isEmbedding_of_restrict_nhds_zero
+ LinearMap.isInducing_of_restrict_nhds_zero
+ smul_tendsto_zero_of_norm_left_le_one

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 175663f).

• +6 new declarations
• −0 removed declarations

+ContinuousSMul.topology_eq_of_induced_eq
+ContinuousSMul.topology_eq_of_nhds_inf_principal_eq
+Filter.IsBoundedUnder.smul_tendsto_zero_normedSpace
+Filter.comap_inf_congr
+LinearMap.isEmbedding_of_restrict_nhds_zero
+LinearMap.isInducing_of_restrict_nhds_zero

---

No changes to strong technical debt.

Increase in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
5014	1	exposed public sections

Current commit 175663f091
Reference commit 4851ebd9bf

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[ADedecker]

Finalized version at #41955. I'm keeping this to remember some results I proved on the way and ended up not needing.