toNormedSpaceCore — Mathlib

English

Analogous statement for the right inner product.

Русский

Аналогичное утверждение для правого внутреннего произведения.

LaTeX

$$$\langle x, x\rangle = 0 \iff x = 0$$$

Lean4

/-- Normed space core structure constructed from an `InnerProductSpace.Core` structure -/
theorem toNormedSpaceCore (cd : InnerProductSpace.Core 𝕜 F) : NormedSpace.Core 𝕜 F
    where
  norm_nonneg x := norm_nonneg x
  norm_eq_zero_iff x := norm_eq_zero
  norm_smul c
    x := by
    letI : NormedSpace 𝕜 F := toNormedSpace
    exact _root_.norm_smul c x
  norm_triangle x y := norm_add_le x y

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