Download ANALYSIS ON GROUPS: WEAK TOPOLOGIES

ANALYSIS ON GROUPS:
WEAK TOPOLOGIES
Recall that a topological vector space is, formally, a pair (E, S ), but that we typically just refer to E
when no confusion can arise from this. In this note, confusion can arise! (But we still refer to just E
as the topological vector space.) For completeness, some of the following exercises are stated in slightly
greater generality than what is strictly needed. If you want to simplify things a bit, try to consider in
general the situation where every topological vector space is a Banach space, and in particular every
dual space is the dual of a Banach space.
Denition 1
(weak topologies). Let E be a vector space, let F be a topological vector space, and let
V be a set of linear maps E → F . The weak topology on E induced by V (the 'V -weak topology' for
short) is dened as the weakest topology S on E such that ϕ is continuous for every ϕ ∈ V . That is,
S = ∩T T , over all topologies such that T on E such that ϕ is T -continuous for every ϕ ∈ V . The
V -weak topology on E is denoted by σ(E, V ).
Exercise 2.
Observe that the V -weak topology is always well-dened.
(i) Show that for any set V as in the previous denition, σ(E, V ) = σ(E, span(V )). That is, we may
as well always assume that V is a vector subspace (and we generally will implicitly assume this in
the sequel).
(ii) Show that σ(E, V ) is generated by the basis of neighbourhoods V (ε, ξ, ϕ1 , . . . , ϕn ), ε > 0, ξ ∈ E, ϕi ∈
V given by
V (ξ, ϕ1 , . . . , ϕn ) := {η ∈ E | ∀i : |ϕi (ξ − η)| < ε}.
(iii) Equivalently, show that a net (ξi )i∈I ⊆ E converges to ξ ∈ E in the V -weak topology, if and only
if ϕ(ξi ) →i ϕ(ξ) in F for all ϕ ∈ V .
(iv) Conclude that E with the V -weak topology is indeed a (not necessarily Hausdor) topological
vector space. Observe that weak topologies are always locally convex. Precisely when are they
Hausdor?
(v) Let V be a subspace of the space of linear maps ϕ : E → R. Show that if ψ : E → R is a linear map
which is σ(E, V )-continuous, then in fact ψ ∈ V .
By the previous exercise, on any locally convex topological vector space E , the E ∗ -weak topology,
which for brevity is usually just called the weak topology on E , is a Hausdor vector topology on E .
Exercise 3.
Show that a convex set in (E, S ) is closed if and only if it is weakly closed. (That is, if
nad only if it is closed in the weak topology.)
Remark 4.
Let's digress a bit. If you want to be really fancy, here is another way to construct σ(E, V )
in such a way that it is, by construction, a vector space topology. Let Ef in be the system of all nite
dimensional subspaces of E . We can then dene a vector space topology Smax on E as the strongest
(locally convex) vector space topology such that the embedding ιF : F → E is continuous for every
F ∈ Ef in .
It can be shown that this is well-dened, and that every linear map ϕ : E → E 0 is Smax -continuous
for any topological vector space E 0 . Hence in the denition of weak topology, we can in fact restrict to
taking the intersection just over (locally convex) vector space topologies.
Let E be a Banach space, E ∗ its continuous dual (that is, E ∗ := {ϕ : E → R | continuous, linear}) with
the norm topology, and let E ∗∗ := (E ∗ )∗ be the double dual of E .
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ANALYSIS ON GROUPS: WEAK TOPOLOGIES
Exercise 5.
Consider the map ι : E → E ∗∗ given by ι(ξ)(ϕ) = ϕ(ξ), ξ ∈ E, ϕ ∈ E ∗ . Show that ι is an
isometric linear map.
Then on E ∗ we get in particular two weak topologies:
(i) The σ(E ∗ , E ∗∗ )-topology. That is, the usual weak topology on E ∗ .
(ii) The σ(E ∗ , ι(E))-topology. This is weaker than the weak topology. It is called the weak-∗ topology
on E ∗ .
Exercise 6. Show that ι(E) is weak-∗ dense in E ∗∗ .
Remark 7. Note that the denition of the weak-∗ topology makes sane on the dual of any topological
vector space.
Let F, F be Banach spaces and T : E → F be a continuous linear map. Then we can dene a
continuous linear map T ∗ : F ∗ → E ∗ by
(T ∗ ϕ)(ξ) = ϕ(T ξ),
ϕ ∈ F ∗ , ξ ∈ E.
Exercise 8.
Show that a norm-continuous linear map S : F ∗ → E ∗ of dual Banach spaces, is continuous
with respect to the weak-∗ topologies if and only if there exists a continuous T : E → F such that
S = T ∗.
HDP, February 17, 2015