100
LECTURE
NOTES
FOR
18.102,
SPRING
2009
Lecture
17.
Thursday
April
9
was
the
second
test
(1)
Problem
1
Let
H
be
a
separable
(partly
because
that
is
mostly
what
I
have
been
talking
about)
Hilbert
space
with
inner
product
(
)
and
norm
·
.
·
,
·
Say
that
a
sequence
u
n
in
H
converges
weakly
if
(
u
n
,v
)
is
Cauchy
in
C
for
each
v
∈
H.
(a)
Explain
why
the
sequence
u
n H
is
bounded.
Solution:
Each
u
n
deFnes
a
continuous
linear
functional
on
H
by
(17.1)
T
n
(
v
) = (
v,u
n
)
,
T
n
=
u
n
,T
n
:
H
−→
C
.
±or
Fxed
v
the
sequence
T
n
(
v
)
is
Cauchy,
and
hence
bounded,
in
C
so
by
the
‘Uniform
Boundedness
Principle’
the
T
n
are
bounded,
hence
u
n
is
bounded
in
R
.
(b)
Show
that
there
exists
an
element
u
∈
H
such
that
(
u
n
,v
)
→
(
u,v
)
for
each
v
∈
H.
Solution:
Since
(
v,u
n
)
is
Cauchy
in
C
for
each
Fxed
v
∈
H
it
is
convergent.
Set
(17.2)
Tv
=
lim
(
v,u
n
)
in
C
.
n
→∞
This
is
a
linear
map,
since
(17.3)
T
(
c
1
v
1
+
c
2
v
2
)
=
lim
c
1
(
v
1
,u
n
) +
c
2
(
v
2
,u
) =
c
1
Tv
1
+
c
2
Tv
2
n
→∞
and
is
bounded
since

Tv
 ≤
C v , C
=
sup
n
u
n
.
Thus,
by
Riesz’
theorem
there
exists
u
∈
H
such
that
Tv
= (
v,u
)
.
Then,
by
deFnition
of
T,
(17.4)
(
u
n
,v
)
→
(
u,v
)
∀
v
∈
H.
(c)
If
e
i
, i
∈
N
,
is
an
orthonormal
sequence,
give,
with
justiFcation,
an
example
of
a
sequence
u
n
which
is
not
weakly
convergent
in
H
but
is
such
that
(
u
n
,e
j
)
converges
for
each
j.
Solution: