norm_fst_le — Mathlib

English

The operator norm of f.prod g equals the norm of the pair (f, g): ||f.prod g|| = ||(f, g)||.

Русский

Норма оператора для произведения f.prod g равна норме пары (f, g): ||f.prod g|| = ||(f, g)||.

LaTeX

$$$$\\|f.\\mathrm{prod}~g\\| = \\|(f, g)\\|.$$$$

Lean4

/-- The operator norm of the first projection `E × F → E` is at most 1. (It is 0 if `E` is zero, so
the inequality cannot be improved without further assumptions.) -/
theorem norm_fst_le : ‖fst 𝕜 E F‖ ≤ 1 :=
  opNorm_le_bound _ zero_le_one (fun ⟨e, f⟩ ↦ by simpa only [one_mul] using le_max_left ‖e‖ ‖f‖)

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