Analysis 2
Homework 02
1. Let
H
be a real Hilbert space on which
φ
is a bounded linear functional;
deﬁne
f
:
H
→
R
:
x
7→ k
x
k
2
+
φ
(
x
)
.
Prove that each (nonempty) closed convex
K
⊂
H
contains a unique point
at which the restriction
f

K
is minimized.
Solution
1. Recall from class the theorem that each closed convex set
K
in
H
contains
a unique point of minimal norm.
We could adopt its
method of proof
. Note ﬁrst that if
x,y
∈
K
then
(parallelogram)
1
2
k
x

y
k
2
=
k
x
k
2
+
k
y
k
2

2
k
1
2
(
x
+
y
)
k
2
whence rewriting squared norms in terms of
f
and
φ
(which cancels by lin
earity) yields
1
2
k
x

y
k
2
=
f
(
x
) +
f
(
y
)

2
f
(
1
2
(
x
+
y
))
.
Now, let
δ
= inf
{
f
(
z
) :
z
∈
K
}
and for each
n
choose
z
n
∈
K
such that (
δ
6
)
f
(
z
n
)
< δ
+1
/n
. As
1
2
(
x
+
y
)
∈
K
it follows from above that if
p,q > n
then
1
2
k
z
p

z
q
k
2
<
(
δ
+ 1
/p
) + (
δ
+ 1
/q
)

2
δ <
2
/n
thus (
z
n
:
n
>
1) is Cauchy and so converges .
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 Spring '09
 LARSON
 Vector Space, Hilbert space, Cauchy, Topological vector space, unique point, real Hilbert space

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