FUNCTIONAL ANALYSIS LECTURE NOTES:
WEAK AND WEAK* CONVERGENCE
CHRISTOPHER HEIL
1.
Weak and Weak* Convergence of Vectors
Definition 1.1.
Let
X
be a normed linear space, and let
x
n
,
x
∈
X
.
a.
We say that
x
n
converges
,
converges strongly
, or
converges in norm
to
x
, and write
x
n
→
x
, if
lim
n
→∞
bardbl
x
−
x
n
bardbl
= 0
.
b. We say that
x
n
converges weakly
to
x
, and write
x
n
w
→
x
, if
∀
μ
∈
X
∗
,
lim
n
→∞
(
x
n
,μ
)
=
(
x,μ
)
.
Exercise 1.2.
a. Show that strong convergence implies weak convergence.
b. Show that weak convergence does not imply strong convergence in general (look for a
Hilbert space counterexample).
If our space is itself the dual space of another space, then there is an additional mode of
convergence that we can consider, as follows.
Definition 1.3.
Let
X
be a normed linear space, and suppose that
μ
n
,
μ
∈
X
∗
. Then we
say that
μ
n
converges weak*
to
μ
, and write
μ
n
w*
−→
μ
, if
∀
x
∈
X,
lim
n
→∞
(
x,μ
n
)
=
(
x,μ
)
.
Note that weak* convergence is just “pointwise convergence” of the operators
μ
n
!
Remark 1.4.
Weak* convergence only makes sense for a sequence that lies in a dual
space
X
∗
. However, if we do have a sequence
{
μ
n
}
n
∈
N
in
X
∗
, then we can consider three
types of convergence of
μ
n
to
μ
: strong, weak, and weak*. By definition, these are:
μ
n
→
μ
⇐⇒
lim
n
→∞
bardbl
μ
−
μ
n
bardbl
= 0
,
μ
n
w
→
μ
⇐⇒
∀
T
∈
X
∗∗
,
lim
n
→∞
(
μ
n
,T
)
=
(
μ,T
)
,
μ
n
w*
−→
μ
⇐⇒
∀
x
∈
X,
lim
n
→∞
(
x,μ
n
)
=
(
x,μ
)
.
c
circlecopyrt
2007 by Christopher Heil.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
WEAK AND WEAK* CONVERGENCE
Exercise 1.5.
Given
μ
n
,
μ
∈
X
∗
, show that
μ
n
→
μ
=
⇒
μ
n
w
→
μ
=
⇒
μ
n
w*
−→
μ.
(1.1)
If
X
is reflexive, show that
μ
n
w
→
μ
⇐⇒
μ
n
w*
−→
μ.
In general, however, the implications in (1.1) do not hold in the reverse direction.
Lemma 1.6.
a. Weak* limits are unique.
b. Weak limits are unique.
Proof.
Suppose that
X
is a normed linear space, and that we had both
μ
n
w*
−→
μ
and
μ
n
w*
−→
ν
in
X
∗
. Then, by definition,
∀
x
∈
X,
(
x,μ
)
=
lim
n
→∞
(
x,μ
n
)
=
(
x,ν
)
,
so
μ
=
ν
.
b. Suppose that we have both
x
n
w
→
x
and
x
n
w
→
y
in
X
. Then, by definition,
∀
μ
∈
X
∗
,
(
x,μ
)
=
lim
n
→∞
(
x
n
,μ
)
=
(
y,μ
)
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '08
 Staff
 Vectors, Convergence, Hilbert space, lim µn, weak* convergence

Click to edit the document details