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

full_name: ContinuousLinearMap.mul
code: /-- Multiplication in a non-unital normed algebra as a continuous bilinear map. -/
def mul : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' 
goal: 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜'

full_name: ContinuousLinearMap.mul_apply'
code: @[simp]
theorem mul_apply' (x y : 𝕜') : mul 𝕜 𝕜' x y = x * y 
goal: ((ContinuousLinearMap.mul 𝕜 𝕜') x) y = x * y

full_name: ContinuousLinearMap.opNorm_mul_apply_le
code: @[simp]
theorem opNorm_mul_apply_le (x : 𝕜') : ‖mul 𝕜 𝕜' x‖ ≤ ‖x‖ 
goal: ‖(ContinuousLinearMap.mul 𝕜 𝕜') x‖ ≤ ‖x‖

full_name: ContinuousLinearMap.opNorm_mul_le
code: theorem opNorm_mul_le : ‖mul 𝕜 𝕜'‖ ≤ 1 
goal: ‖ContinuousLinearMap.mul 𝕜 𝕜'‖ ≤ 1

full_name: NonUnitalAlgHom.Lmul
code: /-- Multiplication on the left in a non-unital normed algebra `𝕜'` as a non-unital algebra
homomorphism into the algebra of *continuous* linear maps. This is the left regular representation
of `A` acting on itself.

This has more algebraic structure than `ContinuousLinearMap.mul`, but there is no longer continuity
bundled in the first coordinate.  An alternative viewpoint is that this upgrades
`NonUnitalAlgHom.lmul` from a homomorphism into linear maps to a homomorphism into *continuous*
linear maps. -/
def _root_.NonUnitalAlgHom.Lmul : 𝕜' →ₙₐ[𝕜] 𝕜' →L[𝕜] 𝕜' 
goal: 𝕜' →ₙₐ[𝕜] 𝕜' →L[𝕜] 𝕜'

full_name: NonUnitalAlgHom.coe_Lmul
code: @[simp]
theorem _root_.NonUnitalAlgHom.coe_Lmul : ⇑(NonUnitalAlgHom.Lmul 𝕜 𝕜') = mul 𝕜 𝕜' 
goal: ⇑(NonUnitalAlgHom.Lmul 𝕜 𝕜') = ⇑(ContinuousLinearMap.mul 𝕜 𝕜')

full_name: ContinuousLinearMap.mulLeftRight
code: /-- Simultaneous left- and right-multiplication in a non-unital normed algebra, considered as a
continuous trilinear map. This is akin to its non-continuous version `LinearMap.mulLeftRight`,
but there is a minor difference: `LinearMap.mulLeftRight` is uncurried. -/
def mulLeftRight : 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' 
goal: 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜' →L[𝕜] 𝕜'

full_name: ContinuousLinearMap.mulLeftRight_apply
code: @[simp]
theorem mulLeftRight_apply (x y z : 𝕜') : mulLeftRight 𝕜 𝕜' x y z = x * z * y 
goal: (((ContinuousLinearMap.mulLeftRight 𝕜 𝕜') x) y) z = x * z * y

full_name: ContinuousLinearMap.opNorm_mulLeftRight_apply_apply_le
code: theorem opNorm_mulLeftRight_apply_apply_le (x y : 𝕜') : ‖mulLeftRight 𝕜 𝕜' x y‖ ≤ ‖x‖ * ‖y‖ 
goal: ‖((ContinuousLinearMap.mulLeftRight 𝕜 𝕜') x) y‖ ≤ ‖x‖ * ‖y‖

full_name: ContinuousLinearMap.opNorm_mulLeftRight_apply_le
code: theorem opNorm_mulLeftRight_apply_le (x : 𝕜') : ‖mulLeftRight 𝕜 𝕜' x‖ ≤ ‖x‖ 
goal: ‖(ContinuousLinearMap.mulLeftRight 𝕜 𝕜') x‖ ≤ ‖x‖

full_name: ContinuousLinearMap.opNorm_mulLeftRight_le
code: theorem opNorm_mulLeftRight_le :
    ‖mulLeftRight 𝕜 𝕜'‖ ≤ 1 
goal: ‖ContinuousLinearMap.mulLeftRight 𝕜 𝕜'‖ ≤ 1

full_name: RegularNormedAlgebra
code: /-- This is a mixin class for non-unital normed algebras which states that the left-regular
representation of the algebra on itself is isometric. Every unital normed algebra with `‖1‖ = 1` is
a regular normed algebra (see `NormedAlgebra.instRegularNormedAlgebra`). In addition, so is every
C⋆-algebra, non-unital included (see `CstarRing.instRegularNormedAlgebra`), but there are yet other
examples. Any algebra with an approximate identity (e.g., $$L^1$$) is also regular.

This is a useful class because it gives rise to a nice norm on the unitization; in particular it is
a C⋆-norm when the norm on `A` is a C⋆-norm. -/
class _root_.RegularNormedAlgebra : Prop :=
  /-- The left regular representation of the algebra on itself is an isometry. -/
  isometry_mul' : Isometry (mul 𝕜 𝕜')
goal: Prop

full_name: RegularNormedAlgebra.isometry_mul'
code: /-- This is a mixin class for non-unital normed algebras which states that the left-regular
representation of the algebra on itself is isometric. Every unital normed algebra with `‖1‖ = 1` is
a regular normed algebra (see `NormedAlgebra.instRegularNormedAlgebra`). In addition, so is every
C⋆-algebra, non-unital included (see `CstarRing.instRegularNormedAlgebra`), but there are yet other
examples. Any algebra with an approximate identity (e.g., $$L^1$$) is also regular.

This is a useful class because it gives rise to a nice norm on the unitization; in particular it is
a C⋆-norm when the norm on `A` is a C⋆-norm. -/
class _root_.RegularNormedAlgebra : Prop :=
  /-- The left regular representation of the algebra on itself is an isometry. -/
  isometry_mul' : Isometry (mul 𝕜 𝕜')
goal: Isometry ⇑(ContinuousLinearMap.mul 𝕜 𝕜')

full_name: NormedAlgebra.instRegularNormedAlgebra
code: /-- Every (unital) normed algebra such that `‖1‖ = 1` is a `RegularNormedAlgebra`. -/
instance _root_.NormedAlgebra.instRegularNormedAlgebra {𝕜 𝕜' : Type*} [NontriviallyNormedField 𝕜]
    [SeminormedRing 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormOneClass 𝕜'] : RegularNormedAlgebra 𝕜 𝕜' 
goal: RegularNormedAlgebra 𝕜 𝕜'

full_name: ContinuousLinearMap.isometry_mul
code: lemma isometry_mul : Isometry (mul 𝕜 𝕜') 
goal: Isometry ⇑(ContinuousLinearMap.mul 𝕜 𝕜')

full_name: ContinuousLinearMap.opNorm_mul_apply
code: @[simp]
lemma opNorm_mul_apply (x : 𝕜') : ‖mul 𝕜 𝕜' x‖ = ‖x‖ 
goal: ‖(ContinuousLinearMap.mul 𝕜 𝕜') x‖ = ‖x‖

full_name: ContinuousLinearMap.opNNNorm_mul_apply
code: @[simp]
lemma opNNNorm_mul_apply (x : 𝕜') : ‖mul 𝕜 𝕜' x‖₊ = ‖x‖₊ 
goal: ‖(ContinuousLinearMap.mul 𝕜 𝕜') x‖₊ = ‖x‖₊

full_name: ContinuousLinearMap.mulₗᵢ
code: /-- Multiplication in a normed algebra as a linear isometry to the space of
continuous linear maps. -/
def mulₗᵢ : 𝕜' →ₗᵢ[𝕜] 𝕜' →L[𝕜] 𝕜' 
goal: 𝕜' →ₗᵢ[𝕜] 𝕜' →L[𝕜] 𝕜'

full_name: ContinuousLinearMap.coe_mulₗᵢ
code: @[simp]
theorem coe_mulₗᵢ : ⇑(mulₗᵢ 𝕜 𝕜') = mul 𝕜 𝕜' 
goal: ⇑(ContinuousLinearMap.mulₗᵢ 𝕜 𝕜') = ⇑(ContinuousLinearMap.mul 𝕜 𝕜')

full_name: ContinuousLinearMap.ring_lmap_equiv_selfₗ
code: /-- If `M` is a normed space over `𝕜`, then the space of maps `𝕜 →L[𝕜] M` is linearly equivalent
to `M`. (See `ring_lmap_equiv_self` for a stronger statement.) -/
def ring_lmap_equiv_selfₗ : (𝕜 →L[𝕜] E) ≃ₗ[𝕜] E 
goal: (𝕜 →L[𝕜] E) ≃ₗ[𝕜] E

full_name: ContinuousLinearMap.ring_lmap_equiv_self
code: /-- If `M` is a normed space over `𝕜`, then the space of maps `𝕜 →L[𝕜] M` is linearly isometrically
equivalent to `M`. -/
def ring_lmap_equiv_self : (𝕜 →L[𝕜] E) ≃ₗᵢ[𝕜] E 
goal: (𝕜 →L[𝕜] E) ≃ₗᵢ[𝕜] E

full_name: ContinuousLinearMap.lsmul
code: /-- Scalar multiplication as a continuous bilinear map. -/
def lsmul : 𝕜' →L[𝕜] E →L[𝕜] E 
goal: 𝕜' →L[𝕜] E →L[𝕜] E

full_name: ContinuousLinearMap.lsmul_apply
code: @[simp]
theorem lsmul_apply (c : 𝕜') (x : E) : lsmul 𝕜 𝕜' c x = c • x 
goal: ((ContinuousLinearMap.lsmul 𝕜 𝕜') c) x = c • x

full_name: ContinuousLinearMap.norm_toSpanSingleton
code: theorem norm_toSpanSingleton (x : E) : ‖toSpanSingleton 𝕜 x‖ = ‖x‖ 
goal: ‖ContinuousLinearMap.toSpanSingleton 𝕜 x‖ = ‖x‖

full_name: ContinuousLinearMap.opNorm_lsmul_apply_le
code: theorem opNorm_lsmul_apply_le (x : 𝕜') : ‖(lsmul 𝕜 𝕜' x : E →L[𝕜] E)‖ ≤ ‖x‖ 
goal: ‖(ContinuousLinearMap.lsmul 𝕜 𝕜') x‖ ≤ ‖x‖

full_name: ContinuousLinearMap.opNorm_lsmul_le
code: /-- The norm of `lsmul` is at most 1 in any semi-normed group. -/
theorem opNorm_lsmul_le : ‖(lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] E →L[𝕜] E)‖ ≤ 1 
goal: ‖ContinuousLinearMap.lsmul 𝕜 𝕜'‖ ≤ 1

full_name: ContinuousLinearMap.opNorm_mul
code: @[simp]
theorem opNorm_mul : ‖mul 𝕜 𝕜'‖ = 1 
goal: ‖ContinuousLinearMap.mul 𝕜 𝕜'‖ = 1

full_name: ContinuousLinearMap.opNNNorm_mul
code: @[simp]
theorem opNNNorm_mul : ‖mul 𝕜 𝕜'‖₊ = 1 
goal: ‖ContinuousLinearMap.mul 𝕜 𝕜'‖₊ = 1

full_name: ContinuousLinearMap.opNorm_lsmul
code: /-- The norm of `lsmul` equals 1 in any nontrivial normed group.

This is `ContinuousLinearMap.opNorm_lsmul_le` as an equality. -/
@[simp]
theorem opNorm_lsmul [NormedField 𝕜'] [NormedAlgebra 𝕜 𝕜'] [NormedSpace 𝕜' E]
    [IsScalarTower 𝕜 𝕜' E] [Nontrivial E] : ‖(lsmul 𝕜 𝕜' : 𝕜' →L[𝕜] E →L[𝕜] E)‖ = 1 
goal: ‖ContinuousLinearMap.lsmul 𝕜 𝕜'‖ = 1