Serge Ballif
MATH 501 Homework 6
November 9, 2007
1.
Let
(
V,
h·
,
·i
)
be an inner product space. Prove that if
h
x,y
i
=
k
x
k·k
y
k
and
y
6
= 0
, then
x
=
ay
for some
a
∈
C
.
We follow the proof outlined in Folland. Deﬁne
a
=
h
x,y
i
h
x,y
i
and
z
=
ay
.
Claim 1.
h
x,z
i
=
h
z,x
i
=
h
x,y
i
and
k
z
k
=
k
y
k
.
Proof.
We use the fact that for
α
∈
C
,
α
α
=

α

2
.
h
x,z
i
=
h
x,ay
i
=
a
h
x,y
i
=
h
y,x
i
h
x,y
i
h
x,y
i
=
h
x,y
i
2
h
x,y
i
=
h
x,y
i
.
h
z,x
i
=
h
ay,x
i
=
a
h
y,x
i
=
h
x,y
i
h
x,y
i
h
y,x
i
=
h
x,y
i
2
h
x,y
i
=
h
x,y
i
.
h
z,z
i
=
h
ay,ay
i
=
a
a
h
y,y
i
=
h
x,y
i
h
x,y
i
h
y,x
i
h
x,y
i
h
y,y
i
=
h
y,y
i
.
Now we can use these facts to show that for
t
∈
R
0
≤ h
x

tz,x

tz
i
=
k
x
k
2
+ 2 Re
h
x,

tz
i
+
k 
tz
k
2
=
k
x
k
2

2
t
h
x,z
i
+
t
2
k
z
k
2
=
k
x
k
2

2
t
h
x,y
i
+
t
2
k
y
k
2
.
Since any quadratic function
ax
2
+
bx
+
c
has its extremum at
x
=

b
2
a
, our
quadratic function in
t
has an absolute minimum at
t
=
h
x,y
i
k
y
k
2
. Substituting this
value in for
t
yields
0
≤ k
x

tz
k
2
=
k
x
k
2

2
t
h
x,y
i
+
t
2
k
y
k
2
=
k
x
k
2

2
h
x,y
i
k
y
k
2
h
x,y
i
+
h
x,y
i
k
y
k
2
2
k
y
k
2
=
k
x
k
2

h
x,y
i
k
y
k
2
.
In this case
h
x,y
i ≤ k
x
kk
y
k
. We get equality in this statement iﬀ
x

tz
=
x

tay
= 0 iﬀ
x
and
y
are linearly dependent.
2.
(a) Let
y
be a nonzero element of a Hilbert space
H
. Prove that, for every
x
∈
H
,
d
(
x,
{
y
}
⊥
) =
h
x,y
i
k
y
k
.
(b) Consider the space
L
2
([0
,
1])
and
E
=
{
f
∈
L
2
([0
,
1])

R
[0
,
1]
f
(
x
)
dm
= 0
}
. Determine
E
⊥
and ﬁnd the distance
of the function
e
x
to
E
.
Page 1 of 5
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentSerge Ballif
MATH 501 Homework 6
November 9, 2007
(a) We know that there exists an element
t
of
{
y
}
⊥
such that
d
(
x,t
) =
d
(
x,
{
y
}
⊥
).
We want to show that
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 WYSOCKI,KRZYSZTOF
 Math, Topology, Rational number, Hilbert space, inner product, Serge Ballif, space H. Prove

Click to edit the document details