File: Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean

full_name: ContinuousLinearMap.nnnorm_def
code: theorem nnnorm_def (f : E →SL[σ₁₂] F) : ‖f‖₊ = sInf { c | ∀ x, ‖f x‖₊ ≤ c * ‖x‖₊ } 
goal: ‖f‖₊ = sInf {c | ∀ (x : E), ‖f x‖₊ ≤ c * ‖x‖₊}

full_name: ContinuousLinearMap.opNNNorm_le_bound
code: /-- If one controls the norm of every `A x`, then one controls the norm of `A`. -/
theorem opNNNorm_le_bound (f : E →SL[σ₁₂] F) (M : ℝ≥0) (hM : ∀ x, ‖f x‖₊ ≤ M * ‖x‖₊) : ‖f‖₊ ≤ M 
goal: ‖f‖₊ ≤ M

full_name: ContinuousLinearMap.opNNNorm_le_bound'
code: /-- If one controls the norm of every `A x`, `‖x‖₊ ≠ 0`, then one controls the norm of `A`. -/
theorem opNNNorm_le_bound' (f : E →SL[σ₁₂] F) (M : ℝ≥0) (hM : ∀ x, ‖x‖₊ ≠ 0 → ‖f x‖₊ ≤ M * ‖x‖₊) :
    ‖f‖₊ ≤ M 
goal: ‖f‖₊ ≤ M

full_name: ContinuousLinearMap.opNNNorm_le_of_unit_nnnorm
code: /-- 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 [NormedSpace ℝ E] [NormedSpace ℝ F] {f : E →L[ℝ] F} {C : ℝ≥0}
    (hf : ∀ x, ‖x‖₊ = 1 → ‖f x‖₊ ≤ C) : ‖f‖₊ ≤ C 
goal: ‖f‖₊ ≤ C

full_name: ContinuousLinearMap.opNNNorm_le_of_lipschitz
code: theorem opNNNorm_le_of_lipschitz {f : E →SL[σ₁₂] F} {K : ℝ≥0} (hf : LipschitzWith K f) :
    ‖f‖₊ ≤ K 
goal: ‖f‖₊ ≤ K

full_name: ContinuousLinearMap.opNNNorm_eq_of_bounds
code: theorem opNNNorm_eq_of_bounds {φ : E →SL[σ₁₂] F} (M : ℝ≥0) (h_above : ∀ x, ‖φ x‖₊ ≤ M * ‖x‖₊)
    (h_below : ∀ N, (∀ x, ‖φ x‖₊ ≤ N * ‖x‖₊) → M ≤ N) : ‖φ‖₊ = M 
goal: ‖φ‖₊ = M

full_name: ContinuousLinearMap.opNNNorm_le_iff
code: theorem opNNNorm_le_iff {f : E →SL[σ₁₂] F} {C : ℝ≥0} : ‖f‖₊ ≤ C ↔ ∀ x, ‖f x‖₊ ≤ C * ‖x‖₊ 
goal: ‖f‖₊ ≤ C ↔ ∀ (x : E), ‖f x‖₊ ≤ C * ‖x‖₊

full_name: ContinuousLinearMap.isLeast_opNNNorm
code: theorem isLeast_opNNNorm : IsLeast {C : ℝ≥0 | ∀ x, ‖f x‖₊ ≤ C * ‖x‖₊} ‖f‖₊ 
goal: IsLeast {C | ∀ (x : E), ‖f x‖₊ ≤ C * ‖x‖₊} ‖f‖₊

full_name: ContinuousLinearMap.opNNNorm_comp_le
code: theorem opNNNorm_comp_le [RingHomIsometric σ₁₃] (f : E →SL[σ₁₂] F) : ‖h.comp f‖₊ ≤ ‖h‖₊ * ‖f‖₊ 
goal: ‖h.comp f‖₊ ≤ ‖h‖₊ * ‖f‖₊

full_name: ContinuousLinearMap.le_opNNNorm
code: theorem le_opNNNorm : ‖f x‖₊ ≤ ‖f‖₊ * ‖x‖₊ 
goal: ‖f x‖₊ ≤ ‖f‖₊ * ‖x‖₊

full_name: ContinuousLinearMap.nndist_le_opNNNorm
code: theorem nndist_le_opNNNorm (x y : E) : nndist (f x) (f y) ≤ ‖f‖₊ * nndist x y 
goal: nndist (f x) (f y) ≤ ‖f‖₊ * nndist x y

full_name: ContinuousLinearMap.lipschitz
code: /-- continuous linear maps are Lipschitz continuous. -/
theorem lipschitz : LipschitzWith ‖f‖₊ f 
goal: LipschitzWith ‖f‖₊ ⇑f

full_name: ContinuousLinearMap.lipschitz_apply
code: /-- Evaluation of a continuous linear map `f` at a point is Lipschitz continuous in `f`. -/
theorem lipschitz_apply (x : E) : LipschitzWith ‖x‖₊ fun f : E →SL[σ₁₂] F => f x 
goal: LipschitzWith ‖x‖₊ fun f => f x

full_name: ContinuousLinearMap.exists_mul_lt_apply_of_lt_opNNNorm
code: theorem exists_mul_lt_apply_of_lt_opNNNorm (f : E →SL[σ₁₂] F) {r : ℝ≥0} (hr : r < ‖f‖₊) :
    ∃ x, r * ‖x‖₊ < ‖f x‖₊ 
goal: ∃ x, r * ‖x‖₊ < ‖f x‖₊

full_name: ContinuousLinearMap.exists_mul_lt_of_lt_opNorm
code: theorem exists_mul_lt_of_lt_opNorm (f : E →SL[σ₁₂] F) {r : ℝ} (hr₀ : 0 ≤ r) (hr : r < ‖f‖) :
    ∃ x, r * ‖x‖ < ‖f x‖ 
goal: ∃ x, r * ‖x‖ < ‖f x‖

full_name: ContinuousLinearMap.exists_lt_apply_of_lt_opNNNorm
code: theorem exists_lt_apply_of_lt_opNNNorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {r : ℝ≥0}
    (hr : r < ‖f‖₊) : ∃ x : E, ‖x‖₊ < 1 ∧ r < ‖f x‖₊ 
goal: ∃ x, ‖x‖₊ < 1 ∧ r < ‖f x‖₊

full_name: ContinuousLinearMap.exists_lt_apply_of_lt_opNorm
code: theorem exists_lt_apply_of_lt_opNorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) {r : ℝ}
    (hr : r < ‖f‖) : ∃ x : E, ‖x‖ < 1 ∧ r < ‖f x‖ 
goal: ∃ x, ‖x‖ < 1 ∧ r < ‖f x‖

full_name: ContinuousLinearMap.sSup_unit_ball_eq_nnnorm
code: theorem sSup_unit_ball_eq_nnnorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
    sSup ((fun x => ‖f x‖₊) '' ball 0 1) = ‖f‖₊ 
goal: sSup ((fun x => ‖f x‖₊) '' Metric.ball 0 1) = ‖f‖₊

full_name: ContinuousLinearMap.sSup_unit_ball_eq_norm
code: theorem sSup_unit_ball_eq_norm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E] [SeminormedAddCommGroup F]
    [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂} [NormedSpace 𝕜 E]
    [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
    sSup ((fun x => ‖f x‖) '' ball 0 1) = ‖f‖ 
goal: sSup ((fun x => ‖f x‖) '' Metric.ball 0 1) = ‖f‖

full_name: ContinuousLinearMap.sSup_closed_unit_ball_eq_nnnorm
code: theorem sSup_closed_unit_ball_eq_nnnorm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
    sSup ((fun x => ‖f x‖₊) '' closedBall 0 1) = ‖f‖₊ 
goal: sSup ((fun x => ‖f x‖₊) '' Metric.closedBall 0 1) = ‖f‖₊

full_name: ContinuousLinearMap.sSup_closed_unit_ball_eq_norm
code: theorem sSup_closed_unit_ball_eq_norm {𝕜 𝕜₂ E F : Type*} [NormedAddCommGroup E]
    [SeminormedAddCommGroup F] [DenselyNormedField 𝕜] [NontriviallyNormedField 𝕜₂] {σ₁₂ : 𝕜 →+* 𝕜₂}
    [NormedSpace 𝕜 E] [NormedSpace 𝕜₂ F] [RingHomIsometric σ₁₂] (f : E →SL[σ₁₂] F) :
    sSup ((fun x => ‖f x‖) '' closedBall 0 1) = ‖f‖ 
goal: sSup ((fun x => ‖f x‖) '' Metric.closedBall 0 1) = ‖f‖

full_name: ContinuousLinearEquiv.lipschitz
code: protected theorem lipschitz : LipschitzWith ‖(e : E →SL[σ₁₂] F)‖₊ e 
goal: LipschitzWith ‖↑e‖₊ ⇑e