nonempty_innerProductSpace — Mathlib

English

There exists an inner product on E making it an inner product space over 𝕜.

Русский

Существует внутреннее произведение на E, делающeE пространством сINNER 𝕜.

LaTeX

$$$\text{nonempty inner product space}$$$

Lean4

/-- **Fréchet–von Neumann–Jordan Theorem**. A normed space `E` whose norm satisfies the
parallelogram identity can be given a compatible inner product. Do
`casesI nonempty_innerProductSpace 𝕜 E` to locally upgrade `InnerProductSpaceable E` to
`InnerProductSpace 𝕜 E`. -/
theorem nonempty_innerProductSpace : Nonempty (InnerProductSpace 𝕜 E) :=
  ⟨{  inner := inner_ 𝕜
      norm_sq_eq_re_inner := inner_.norm_sq
      conj_inner_symm := inner_.conj_symm
      add_left := add_left
      smul_left := fun _ _ _ => innerProp _ _ _ }⟩

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