- #1
HMY
- 14
- 0
Let LG be the base point preserving loops (it's a Hilbert manifold).
So LG = { f : S^1 -> G s.t. f(0)=1 } where G is a connected, simply connected Lie group.
LG is embedded into the (vector space) Hilbert space L^2[0, 2pi]
given by f |--> g(t) = f '(t)f(t)^-1
Is LG a closed subset of L^2[0,2pi] with respect to the weak topology?
(or is the embedding weakly continuous)?
I figured out the weak topology (reminder here):
Let X = L^2[0,2pi]
Let U(F,b) := { x in X | |F(x)| < b } where b is in R & F is in X^*
So {U(F,b)} are a basis of a neighbourhood of 0 in X.
Thus {x + U(F,b) } are a basis of a neighbourhood of x in X.
ie. the neighbourhoods of an arbitrary x in X are precisely the translates x + W of
neighbourhoods W of 0
So the weak topology on LG is just the subset topology coming from the weak topology
defined on X.
So LG = { f : S^1 -> G s.t. f(0)=1 } where G is a connected, simply connected Lie group.
LG is embedded into the (vector space) Hilbert space L^2[0, 2pi]
given by f |--> g(t) = f '(t)f(t)^-1
Is LG a closed subset of L^2[0,2pi] with respect to the weak topology?
(or is the embedding weakly continuous)?
I figured out the weak topology (reminder here):
Let X = L^2[0,2pi]
Let U(F,b) := { x in X | |F(x)| < b } where b is in R & F is in X^*
So {U(F,b)} are a basis of a neighbourhood of 0 in X.
Thus {x + U(F,b) } are a basis of a neighbourhood of x in X.
ie. the neighbourhoods of an arbitrary x in X are precisely the translates x + W of
neighbourhoods W of 0
So the weak topology on LG is just the subset topology coming from the weak topology
defined on X.