- #1
sponsoredwalk
- 533
- 5
I have a question about mappings that go from a vector space to the dual space, the
notation is quite strange.
A linear functional is just a linear map f : V → F.
The dual space of V is the vector space L(V,F) = (V)*, i.e. the space
of linear functionals, i.e. maps from V to F.
L(V,F)= { f : V → F | [f(x + y) = f(x) + f(y)] ⋀ [ f(λx) = λf(x) ]}.
That' supposed to read "The set of all functions from V to F such that f is linear" (i.e. a
linear functional), if you have a better, clearer way to express that please post it!
When you're dealing with a linear transformation from a vector space to the dual space
I am really worried about the form it takes: T : V → L(V,F), i.e. T : V → (V)*.
Reading the start of Spivak's Calculus on Manifolds he mentions this explicitly and more
as below. This is what I find so weird, you're mapping an element of a vector space
to a map, not just some element but to an element of a vector space which
itself is a map, mapping from the vector space to the field...
If (ℝⁿ)* is the dual space to ℝⁿ, with x ∈ ℝⁿ,
define φx ∈ (ℝⁿ)* by φx(y) = <x,y>,
define T : ℝⁿ → (ℝⁿ)* by T(x) = φx. This comes from Spivak's CoM, He's
clearly written that φx ∈ (ℝⁿ)*, but I get the feeling that sometimes
you're working with the scalar value <x,y> also supposed to be in (ℝⁿ)*
& other times not. For example in Susskind's quantum mechanics lectures he explicitly
says that this is all just a way to play with inner products & get complex numbers out,
but again I want to get this rigorous because it's troubling me with regard to differential
forms in particular. But to put it all into my notation of:
T : ℝⁿ → (ℝⁿ)* | x ↦ T(x) = φx.
the element φx is itself a map from ℝⁿ to ℝ so you pick a vector from the
dual space and map it onto the inner product. To spell it all out:
T : ℝⁿ → L(ℝⁿ,ℝ) | x ↦ T(x) = φx : ℝⁿ → ℝ | y ↦ φx(y) = <x,y>.
Here x ∈ ℝⁿ is mapped onto T(x) = φx ∈ (ℝⁿ)* = L(ℝⁿ,ℝ) but this
itself is mapping y ∈ ℝⁿ to <x,y> ∈ ℝ
I'm going to say that this is very weird. For example, does this:
T : ℝⁿ → (ℝⁿ)* | (λx + y) ↦ T(λx + y) = λφx + φy
then
λφx : ℝⁿ → ℝ | z ↦ λφx(z) = λ<x,z>
φy : ℝⁿ → ℝ | z ↦ φy(z) = <y,z>,
or more explicitly,
T : ℝⁿ → (ℝⁿ)* | (λx + y) ↦ T(λx + y) = λφx + φy = [λφx : ℝⁿ → ℝ | z ↦ λφx(z) = λ<x,z>] + [φy : ℝⁿ → ℝ | z ↦ φy(z) = <y,z>] = λ<x,z> + <y,z>
make sense? Clearly here you've got this big chain of map to the value of the map which
itself is a map to the real numbers & I point this out because I see no reason to think it's
wrong. Any thoughts? The notation is the most important thing here but if you would be
able to include some thoughts on how this notation works in quantum mechanics &
differential forms that would be great.
notation is quite strange.
A linear functional is just a linear map f : V → F.
The dual space of V is the vector space L(V,F) = (V)*, i.e. the space
of linear functionals, i.e. maps from V to F.
L(V,F)= { f : V → F | [f(x + y) = f(x) + f(y)] ⋀ [ f(λx) = λf(x) ]}.
That' supposed to read "The set of all functions from V to F such that f is linear" (i.e. a
linear functional), if you have a better, clearer way to express that please post it!
When you're dealing with a linear transformation from a vector space to the dual space
I am really worried about the form it takes: T : V → L(V,F), i.e. T : V → (V)*.
Reading the start of Spivak's Calculus on Manifolds he mentions this explicitly and more
as below. This is what I find so weird, you're mapping an element of a vector space
to a map, not just some element but to an element of a vector space which
itself is a map, mapping from the vector space to the field...
If (ℝⁿ)* is the dual space to ℝⁿ, with x ∈ ℝⁿ,
define φx ∈ (ℝⁿ)* by φx(y) = <x,y>,
define T : ℝⁿ → (ℝⁿ)* by T(x) = φx. This comes from Spivak's CoM, He's
clearly written that φx ∈ (ℝⁿ)*, but I get the feeling that sometimes
you're working with the scalar value <x,y> also supposed to be in (ℝⁿ)*
& other times not. For example in Susskind's quantum mechanics lectures he explicitly
says that this is all just a way to play with inner products & get complex numbers out,
but again I want to get this rigorous because it's troubling me with regard to differential
forms in particular. But to put it all into my notation of:
T : ℝⁿ → (ℝⁿ)* | x ↦ T(x) = φx.
the element φx is itself a map from ℝⁿ to ℝ so you pick a vector from the
dual space and map it onto the inner product. To spell it all out:
T : ℝⁿ → L(ℝⁿ,ℝ) | x ↦ T(x) = φx : ℝⁿ → ℝ | y ↦ φx(y) = <x,y>.
Here x ∈ ℝⁿ is mapped onto T(x) = φx ∈ (ℝⁿ)* = L(ℝⁿ,ℝ) but this
itself is mapping y ∈ ℝⁿ to <x,y> ∈ ℝ
I'm going to say that this is very weird. For example, does this:
T : ℝⁿ → (ℝⁿ)* | (λx + y) ↦ T(λx + y) = λφx + φy
then
λφx : ℝⁿ → ℝ | z ↦ λφx(z) = λ<x,z>
φy : ℝⁿ → ℝ | z ↦ φy(z) = <y,z>,
or more explicitly,
T : ℝⁿ → (ℝⁿ)* | (λx + y) ↦ T(λx + y) = λφx + φy = [λφx : ℝⁿ → ℝ | z ↦ λφx(z) = λ<x,z>] + [φy : ℝⁿ → ℝ | z ↦ φy(z) = <y,z>] = λ<x,z> + <y,z>
make sense? Clearly here you've got this big chain of map to the value of the map which
itself is a map to the real numbers & I point this out because I see no reason to think it's
wrong. Any thoughts? The notation is the most important thing here but if you would be
able to include some thoughts on how this notation works in quantum mechanics &
differential forms that would be great.
Last edited: