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[Merged by Bors] - refactor(analysis/complex/basic, data/complex/is_R_or_C): downgrade imports#18217
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[Merged by Bors] - refactor(analysis/complex/basic, data/complex/is_R_or_C): downgrade imports#18217
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Half of this file is completely elementary, able to be proved directly from the definitions in `normed/group/hom/basic` after a few instances are added there. The other half consists of technical lemmas from LTE, never used elsewhere in mathlib, and requires more imports. Since this file is imported by many files (including `data/complex/is_R_or_C`, see #18217 for a discussion of what that file imports), I propose splitting off the LTE half.
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…mports (#18217) The files `data/complex/is_R_or_C` and `analysis/complex/basic` are imported widely across the analysis library: for example - `data/complex/is_R_or_C` into inner product spaces - `analysis/complex/basic` into the construction of `rpow` and thence into Lp spaces and the Bochner integral So it is useful (for the port and for compilation time) to make them as elementary as possible. Currently they both import `analysis/normed_space/operator_norm` and `analysis/normed_space/finite_dimension`, rather heavy imports, but use quite little from these files: they provide lemmas about the operator norms of `re`/`im`/`conj`/`of_real`, and get cheaply some facts about the topology of `ℂ`/`is_R_or_C` via real-finite-dimensionality. This PR splits both files. - `analysis/complex/basic` is split in place, with substantially downgraded imports, and with a few heavier lemmas moved to `analysis/complex/operator_norm`. (And a few lemmas moved earlier to `data/complex/module`.). I wrote direct proofs for the completeness and properness of `ℂ` so that these facts can remain available by importing this file, even though with heavier imports these facts can be deduced from the real-finite-dimensionality of `ℂ`. - `data/complex/is_R_or_C` is split into `data/is_R_or_C/basic` (lightweight file containing most of the former file) and `data/is_R_or_C/lemmas` (a few heavier lemmas). Also rename `equiv_real_prod_add_hom_lm` to `equiv_real_prod_lm` (an apparent typo), and `equiv_real_prodₗ` to `equiv_real_prod_clm` (also an apparent error since in our naming convention `ₗ` usually refers to linearity, not continuous-linearity).
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…st year; I presume the real code goes back to leanprover-community/mathlib3#1716 in 2019.
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The files
data/complex/is_R_or_Candanalysis/complex/basicare imported widely across the analysis library: for exampledata/complex/is_R_or_Cinto inner product spacesanalysis/complex/basicinto the construction ofrpowand thence into Lp spaces and the Bochner integralSo it is useful (for the port and for compilation time) to make them as elementary as possible.
Currently they both import
analysis/normed_space/operator_normandanalysis/normed_space/finite_dimension, rather heavy imports, but use quite little from these files: they provide lemmas about the operator norms ofre/im/conj/of_real, and get cheaply some facts about the topology ofℂ/is_R_or_Cvia real-finite-dimensionality.This PR splits both files.
analysis/complex/basicis split in place, with substantially downgraded imports, and with a few heavier lemmas moved toanalysis/complex/operator_norm. (And a few lemmas moved earlier todata/complex/module.). I wrote direct proofs for the completeness and properness ofℂso that these facts can remain available by importing this file, even though with heavier imports these facts can be deduced from the real-finite-dimensionality ofℂ.data/complex/is_R_or_Cis split intodata/is_R_or_C/basic(lightweight file containing most of the former file) anddata/is_R_or_C/lemmas(a few heavier lemmas).Also rename
equiv_real_prod_add_hom_lmtoequiv_real_prod_lm(an apparent typo), andequiv_real_prodₗtoequiv_real_prod_clm(also an apparent error since in our naming conventionₗusually refers to linearity, not continuous-linearity).