Let LG be the base point preserving loops (it's a Hilbert manifold).(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# Closed subset (with respect to weak topology)?

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