Continuous multilinear maps
We define continuous multilinear maps as maps from Π(i : ι), M₁ i to M₂ which are multilinear
and continuous, by extending the space of multilinear maps with a continuity assumption.
Here, M₁ i and M₂ are modules over a ring R, and ι is an arbitrary type, and all these
spaces are also topological spaces.
Main definitions
continuous_multilinear_map R M₁ M₂is the space of continuous multilinear maps fromΠ(i : ι), M₁ itoM₂. We show that it is anR-module.
Implementation notes
We mostly follow the API of multilinear maps.
Notation
We introduce the notation M [×n]→L[R] M' for the space of continuous n-multilinear maps from
M^n to M'. This is a particular case of the general notion (where we allow varying dependent
types as the arguments of our continuous multilinear maps), but arguably the most important one,
especially when defining iterated derivatives.
structure
continuous_multilinear_map
(R : Type u)
{ι : Type v}
(M₁ : ι → Type w₁)
(M₂ : Type w₂)
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂] :
Type (max v w₁ w₂)
- to_multilinear_map : multilinear_map R M₁ M₂
- cont : continuous c.to_multilinear_map.to_fun
Continuous multilinear maps over the ring R, from Πi, M₁ i to M₂ where M₁ i and M₂
are modules over R with a topological structure. In applications, there will be compatibility
conditions between the algebraic and the topological structures, but this is not needed for the
definition.
@[instance]
def
continuous_multilinear_map.has_coe_to_fun
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂] :
has_coe_to_fun (continuous_multilinear_map R M₁ M₂)
Equations
- continuous_multilinear_map.has_coe_to_fun = {F := λ (f : continuous_multilinear_map R M₁ M₂), (Π (i : ι), M₁ i) → M₂, coe := λ (f : continuous_multilinear_map R M₁ M₂), f.to_multilinear_map.to_fun}
theorem
continuous_multilinear_map.coe_continuous
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂) :
@[simp]
theorem
continuous_multilinear_map.coe_coe
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂) :
⇑(f.to_multilinear_map) = ⇑f
@[ext]
theorem
continuous_multilinear_map.ext
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
{f f' : continuous_multilinear_map R M₁ M₂} :
@[simp]
theorem
continuous_multilinear_map.map_add
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
(m : Π (i : ι), M₁ i)
(i : ι)
(x y : M₁ i) :
⇑f (function.update m i (x + y)) = ⇑f (function.update m i x) + ⇑f (function.update m i y)
@[simp]
theorem
continuous_multilinear_map.map_smul
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
(m : Π (i : ι), M₁ i)
(i : ι)
(c : R)
(x : M₁ i) :
⇑f (function.update m i (c • x)) = c • ⇑f (function.update m i x)
theorem
continuous_multilinear_map.map_coord_zero
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
{m : Π (i : ι), M₁ i}
(i : ι) :
@[simp]
theorem
continuous_multilinear_map.map_zero
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
[nonempty ι] :
@[instance]
def
continuous_multilinear_map.has_zero
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂] :
has_zero (continuous_multilinear_map R M₁ M₂)
@[instance]
def
continuous_multilinear_map.inhabited
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂] :
inhabited (continuous_multilinear_map R M₁ M₂)
Equations
@[simp]
theorem
continuous_multilinear_map.zero_apply
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(m : Π (i : ι), M₁ i) :
@[instance]
def
continuous_multilinear_map.has_add
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[has_continuous_add M₂] :
has_add (continuous_multilinear_map R M₁ M₂)
Equations
- continuous_multilinear_map.has_add = {add := λ (f f' : continuous_multilinear_map R M₁ M₂), {to_multilinear_map := {to_fun := (f.to_multilinear_map + f'.to_multilinear_map).to_fun, map_add' := _, map_smul' := _}, cont := _}}
@[simp]
theorem
continuous_multilinear_map.add_apply
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f f' : continuous_multilinear_map R M₁ M₂)
[has_continuous_add M₂]
(m : Π (i : ι), M₁ i) :
@[instance]
def
continuous_multilinear_map.add_comm_monoid
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[has_continuous_add M₂] :
add_comm_monoid (continuous_multilinear_map R M₁ M₂)
Equations
- continuous_multilinear_map.add_comm_monoid = {add := has_add.add continuous_multilinear_map.has_add, add_assoc := _, zero := 0, zero_add := _, add_zero := _, add_comm := _}
@[simp]
theorem
continuous_multilinear_map.sum_apply
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[has_continuous_add M₂]
{α : Type u_1}
(f : α → continuous_multilinear_map R M₁ M₂)
(m : Π (i : ι), M₁ i)
{s : finset α} :
def
continuous_multilinear_map.to_continuous_linear_map
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
(m : Π (i : ι), M₁ i)
(i : ι) :
If f is a continuous multilinear map, then f.to_continuous_linear_map m i is the continuous
linear map obtained by fixing all coordinates but i equal to those of m, and varying the
i-th coordinate.
Equations
- f.to_continuous_linear_map m i = {to_linear_map := {to_fun := (f.to_multilinear_map.to_linear_map m i).to_fun, map_add' := _, map_smul' := _}, cont := _}
def
continuous_multilinear_map.prod
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
{M₃ : Type w₃}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[add_comm_monoid M₃]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[semimodule R M₃]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[topological_space M₃] :
continuous_multilinear_map R M₁ M₂ → continuous_multilinear_map R M₁ M₃ → continuous_multilinear_map R M₁ (M₂ × M₃)
The cartesian product of two continuous multilinear maps, as a continuous multilinear map.
Equations
- f.prod g = {to_multilinear_map := {to_fun := (f.to_multilinear_map.prod g.to_multilinear_map).to_fun, map_add' := _, map_smul' := _}, cont := _}
@[simp]
theorem
continuous_multilinear_map.prod_apply
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
{M₃ : Type w₃}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[add_comm_monoid M₃]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[semimodule R M₃]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[topological_space M₃]
(f : continuous_multilinear_map R M₁ M₂)
(g : continuous_multilinear_map R M₁ M₃)
(m : Π (i : ι), M₁ i) :
def
continuous_multilinear_map.comp_continuous_linear_map
(R : Type u)
{ι : Type v}
{M₂ : Type w₂}
(M₃ : Type w₃)
{M₄ : Type w₄}
[decidable_eq ι]
[semiring R]
[add_comm_monoid M₂]
[add_comm_monoid M₃]
[add_comm_monoid M₄]
[semimodule R M₂]
[semimodule R M₃]
[semimodule R M₄]
[topological_space M₂]
[topological_space M₃]
[topological_space M₄] :
continuous_multilinear_map R (λ (i : ι), M₃) M₄ → (M₂ →L[R] M₃) → continuous_multilinear_map R (λ (i : ι), M₂) M₄
If g is continuous multilinear and f is continuous linear, then g (f m₁, ..., f mₙ) is
again a continuous multilinear map, that we call g.comp_continuous_linear_map f.
Equations
- continuous_multilinear_map.comp_continuous_linear_map R M₃ g f = {to_multilinear_map := {to_fun := (multilinear_map.comp_linear_map R M₃ g.to_multilinear_map f.to_linear_map).to_fun, map_add' := _, map_smul' := _}, cont := _}
theorem
continuous_multilinear_map.cons_add
{R : Type u}
{n : ℕ}
{M : fin n.succ → Type w}
{M₂ : Type w₂}
[semiring R]
[Π (i : fin n.succ), add_comm_monoid (M i)]
[add_comm_monoid M₂]
[Π (i : fin n.succ), semimodule R (M i)]
[semimodule R M₂]
[Π (i : fin n.succ), topological_space (M i)]
[topological_space M₂]
(f : continuous_multilinear_map R M M₂)
(m : Π (i : fin n), M i.succ)
(x y : M 0) :
In the specific case of continuous multilinear maps on spaces indexed by fin (n+1), where one
can build an element of Π(i : fin (n+1)), M i using cons, one can express directly the
additivity of a multilinear map along the first variable.
theorem
continuous_multilinear_map.cons_smul
{R : Type u}
{n : ℕ}
{M : fin n.succ → Type w}
{M₂ : Type w₂}
[semiring R]
[Π (i : fin n.succ), add_comm_monoid (M i)]
[add_comm_monoid M₂]
[Π (i : fin n.succ), semimodule R (M i)]
[semimodule R M₂]
[Π (i : fin n.succ), topological_space (M i)]
[topological_space M₂]
(f : continuous_multilinear_map R M M₂)
(m : Π (i : fin n), M i.succ)
(c : R)
(x : M 0) :
In the specific case of continuous multilinear maps on spaces indexed by fin (n+1), where one
can build an element of Π(i : fin (n+1)), M i using cons, one can express directly the
multiplicativity of a multilinear map along the first variable.
theorem
continuous_multilinear_map.map_piecewise_add
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
(m m' : Π (i : ι), M₁ i)
(t : finset ι) :
theorem
continuous_multilinear_map.map_add_univ
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
[fintype ι]
(m m' : Π (i : ι), M₁ i) :
Additivity of a continuous multilinear map along all coordinates at the same time,
writing f (m + m') as the sum of f (s.piecewise m m') over all sets s.
theorem
continuous_multilinear_map.map_sum_finset
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
{α : ι → Type u_1}
[fintype ι]
(g : Π (i : ι), α i → M₁ i)
(A : Π (i : ι), finset (α i)) :
If f is continuous multilinear, then f (Σ_{j₁ ∈ A₁} g₁ j₁, ..., Σ_{jₙ ∈ Aₙ} gₙ jₙ) is the sum
of f (g₁ (r 1), ..., gₙ (r n)) where r ranges over all functions with r 1 ∈ A₁, ...,
r n ∈ Aₙ. This follows from multilinearity by expanding successively with respect to each
coordinate.
theorem
continuous_multilinear_map.map_sum
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[semiring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
{α : ι → Type u_1}
[fintype ι]
(g : Π (i : ι), α i → M₁ i)
[Π (i : ι), fintype (α i)] :
⇑f (λ (i : ι), finset.univ.sum (λ (j : α i), g i j)) = finset.univ.sum (λ (r : Π (i : ι), α i), ⇑f (λ (i : ι), g i (r i)))
If f is continuous multilinear, then f (Σ_{j₁} g₁ j₁, ..., Σ_{jₙ} gₙ jₙ) is the sum of
f (g₁ (r 1), ..., gₙ (r n)) where r ranges over all functions r. This follows from
multilinearity by expanding successively with respect to each coordinate.
@[simp]
theorem
continuous_multilinear_map.map_sub
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[ring R]
[Π (i : ι), add_comm_group (M₁ i)]
[add_comm_group M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
(m : Π (i : ι), M₁ i)
(i : ι)
(x y : M₁ i) :
⇑f (function.update m i (x - y)) = ⇑f (function.update m i x) - ⇑f (function.update m i y)
@[instance]
def
continuous_multilinear_map.has_neg
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[ring R]
[Π (i : ι), add_comm_group (M₁ i)]
[add_comm_group M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[topological_add_group M₂] :
has_neg (continuous_multilinear_map R M₁ M₂)
Equations
- continuous_multilinear_map.has_neg = {neg := λ (f : continuous_multilinear_map R M₁ M₂), {to_multilinear_map := {to_fun := (-f.to_multilinear_map).to_fun, map_add' := _, map_smul' := _}, cont := _}}
@[simp]
theorem
continuous_multilinear_map.neg_apply
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[ring R]
[Π (i : ι), add_comm_group (M₁ i)]
[add_comm_group M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
[topological_add_group M₂]
(m : Π (i : ι), M₁ i) :
@[instance]
def
continuous_multilinear_map.add_comm_group
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[ring R]
[Π (i : ι), add_comm_group (M₁ i)]
[add_comm_group M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[topological_add_group M₂] :
add_comm_group (continuous_multilinear_map R M₁ M₂)
Equations
- continuous_multilinear_map.add_comm_group = {add := has_add.add continuous_multilinear_map.has_add, add_assoc := _, zero := 0, zero_add := _, add_zero := _, neg := has_neg.neg continuous_multilinear_map.has_neg, add_left_neg := _, add_comm := _}
@[simp]
theorem
continuous_multilinear_map.sub_apply
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[ring R]
[Π (i : ι), add_comm_group (M₁ i)]
[add_comm_group M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f f' : continuous_multilinear_map R M₁ M₂)
[topological_add_group M₂]
(m : Π (i : ι), M₁ i) :
theorem
continuous_multilinear_map.map_piecewise_smul
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[comm_ring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
(c : ι → R)
(m : Π (i : ι), M₁ i)
(s : finset ι) :
theorem
continuous_multilinear_map.map_smul_univ
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[comm_ring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
[fintype ι]
(c : ι → R)
(m : Π (i : ι), M₁ i) :
Multiplicativity of a continuous multilinear map along all coordinates at the same time,
writing f (λ i, c i • m i) as (∏ i, c i) • f m.
@[instance]
def
continuous_multilinear_map.has_scalar
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[comm_ring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[topological_space R]
[topological_semimodule R M₂] :
has_scalar R (continuous_multilinear_map R M₁ M₂)
Equations
- continuous_multilinear_map.has_scalar = {smul := λ (c : R) (f : continuous_multilinear_map R M₁ M₂), {to_multilinear_map := {to_fun := (c • f.to_multilinear_map).to_fun, map_add' := _, map_smul' := _}, cont := _}}
@[simp]
theorem
continuous_multilinear_map.smul_apply
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[comm_ring R]
[Π (i : ι), add_comm_monoid (M₁ i)]
[add_comm_monoid M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
(f : continuous_multilinear_map R M₁ M₂)
[topological_space R]
[topological_semimodule R M₂]
(c : R)
(m : Π (i : ι), M₁ i) :
@[instance]
def
continuous_multilinear_map.semimodule
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[comm_ring R]
[Π (i : ι), add_comm_group (M₁ i)]
[add_comm_group M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[topological_add_group M₂]
[topological_space R]
[topological_semimodule R M₂] :
semimodule R (continuous_multilinear_map R M₁ M₂)
The space of continuous multilinear maps is a module over R, for the pointwise addition and
scalar multiplication.
Equations
- continuous_multilinear_map.semimodule = semimodule.of_core {to_has_scalar := {smul := has_scalar.smul continuous_multilinear_map.has_scalar}, smul_add := _, add_smul := _, mul_smul := _, one_smul := _}
def
continuous_multilinear_map.to_multilinear_map_linear
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
[decidable_eq ι]
[comm_ring R]
[Π (i : ι), add_comm_group (M₁ i)]
[add_comm_group M₂]
[Π (i : ι), semimodule R (M₁ i)]
[semimodule R M₂]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[topological_add_group M₂]
[topological_space R]
[topological_semimodule R M₂] :
continuous_multilinear_map R M₁ M₂ →ₗ[R] multilinear_map R M₁ M₂
Linear map version of the map to_multilinear_map associating to a continuous multilinear map
the corresponding multilinear map.
Equations
- continuous_multilinear_map.to_multilinear_map_linear = {to_fun := λ (f : continuous_multilinear_map R M₁ M₂), f.to_multilinear_map, map_add' := _, map_smul' := _}
def
continuous_linear_map.comp_continuous_multilinear_map
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
{M₃ : Type w₃}
[decidable_eq ι]
[ring R]
[Π (i : ι), add_comm_group (M₁ i)]
[add_comm_group M₂]
[add_comm_group M₃]
[Π (i : ι), module R (M₁ i)]
[module R M₂]
[module R M₃]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[topological_space M₃] :
(M₂ →L[R] M₃) → continuous_multilinear_map R M₁ M₂ → continuous_multilinear_map R M₁ M₃
Composing a continuous multilinear map with a continuous linear map gives again a continuous multilinear map.
Equations
- g.comp_continuous_multilinear_map f = {to_multilinear_map := {to_fun := (g.to_linear_map.comp_multilinear_map f.to_multilinear_map).to_fun, map_add' := _, map_smul' := _}, cont := _}
@[simp]
theorem
continuous_linear_map.comp_continuous_multilinear_map_coe
{R : Type u}
{ι : Type v}
{M₁ : ι → Type w₁}
{M₂ : Type w₂}
{M₃ : Type w₃}
[decidable_eq ι]
[ring R]
[Π (i : ι), add_comm_group (M₁ i)]
[add_comm_group M₂]
[add_comm_group M₃]
[Π (i : ι), module R (M₁ i)]
[module R M₂]
[module R M₃]
[Π (i : ι), topological_space (M₁ i)]
[topological_space M₂]
[topological_space M₃]
(g : M₂ →L[R] M₃)
(f : continuous_multilinear_map R M₁ M₂) :