ofReal_tsum — Mathlib

English

For any f: α → ℝ, the canonical embedding commutes with tsum: (ofReal (tsum f L)) = tsum (ofReal ∘ f) L. Equivalently, embedding the total sum equals the sum of embedded terms.

Русский

Пусть f : α → ℝ. Каноническое встраивание в 𝕜 commuting с tsum: ofReal(tsum f L) = tsum (ofReal ∘ f) L.

LaTeX

$$$ \\operatorname{ofReal}\\Bigl( \\sum'_{L} a, f(a) \\Bigr) = \\sum'_{L} a, (\\operatorname{ofReal}(f(a))) $$$

Lean4

@[norm_cast]
theorem ofReal_tsum (f : α → ℝ) : (↑(∑'[L] a, f a) : 𝕜) = ∑'[L] a, (f a : 𝕜) :=
  Function.LeftInverse.map_tsum f ofRealCLM.continuous continuous_re (fun _ ↦ by simp)

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