[Merged by Bors] - chore(Analysis/InnerProductSpace): fix eigenvalue_pos_of_pos

[mcdoll]

The current statement of the lemma eigenvalue_pos_of_pos is non-sense as its assumption can never be satisfied (it is always violated for x=0). Fix this; also remove some type annotations that are completely unnecessary.

---

import Mathlib.Analysis.InnerProductSpace.Spectrum

variable {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]

local notation "⟪" x ", " y "⟫" => inner 𝕜 x y

open Module.End

theorem false_of_pos {μ : ℝ} {T : E →ₗ[𝕜] E} (_hμ : HasEigenvalue T μ)
    (hnn : ∀ x : E, 0 < RCLike.re ⟪x, T x⟫) : False := by
  suffices ¬∀ x : E, 0 < RCLike.re ⟪x, T x⟫ from this hnn
  simp only [not_forall, not_lt]
  use 0
  simp

Also remove some type annotations that are completely unnecessary.

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[github-actions]

PR summary f24777c72c

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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---

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Current commit f24777c72c
Reference commit 818f61df41

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[grunweg]

Good find, thanks! I've shortened your PR description a bit (since right now, the description is longer than the fix). If you disagree with this, feel free to edit it back.
bors d+

[mathlib-bors]

✌️ mcdoll can now approve this pull request until 2026-09-02 08:06 UTC (in 2 weeks). To approve and merge, reply with bors r+. More detailed instructions are available here.

⚠️ This delegation only covers changes within Archive/**, Counterexamples/**, docs/**, DownstreamTest/**, Mathlib/**, MathlibTest/**, Wanted/**, widget/**, Archive.lean, Counterexamples.lean, docs.lean, Mathlib.lean, Wanted.lean; an author commit touching anything else will revoke it. Bors also revokes it if a later push changes too many files for it to check the full list — even if it stays within scope.

[mcdoll]

Thanks, yes the example should not be part of the commit message.
bors merge

[mathlib-bors]

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