Closed subset (with respect to weak topology)?

  1. 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.
  2. jcsd
  3. Nets give a nice characterization of the weak topology...

  4. What would this be?

    I'm not fluent with nets (yet) but if this characterization is nice enough then it is perhaps easier for me to work with it here, than what I was trying before.
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