opNNNorm_le_of_unit_nnnorm — Mathlib

English

If f satisfies an upper bound by M and no smaller N can bound f, then ‖f‖₊ = M.

Русский

Если f удовлетворяет верхней границе M, и никакое меньшее N не может обеспечить bound, тогда ‖f‖₊ = M.

LaTeX

$$$\\|f\\|_{+} = M$ under appropriate upper/lower bound conditions$$

Lean4

/-- For a continuous real linear map `f`, if one controls the norm of every `f x`, `‖x‖₊ = 1`, then
one controls the norm of `f`. -/
theorem opNNNorm_le_of_unit_nnnorm [NormedAlgebra ℝ 𝕜] {f : E →SL[σ₁₂] F} {C : ℝ≥0} (hf : ∀ x, ‖x‖₊ = 1 → ‖f x‖₊ ≤ C) :
    ‖f‖₊ ≤ C :=
  opNorm_le_of_unit_norm C.coe_nonneg fun x hx => hf x <| by rwa [← NNReal.coe_eq_one]

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