Mathematics Department Stanford University
Math 175 Homework 7
Due Thursday May 20.
1. (i) Let
X
and
Y
be Banach spaces and let
B
n
be the open ball around
0
in
X
with radius
n
. Let
M
∈
L
(
X, Y
)
be an onto map, that is,
Range
(
M
) =
Y
.
Argue that at least one of the sets
MB
n
contains an open ball inside.
Hint: use Q4 from HW2.
(ii) Prove that there exists such
r >
0
that
MB
1
contains ball
B
r
(0)
and conse
quently that for all
c >
0
,
the set
MB
c
(0)
contains the ball
B
cr
(0)
.
Hint: Show that for some
x
0
and
B
n
from (i) it is true that
M
(
B
n
+
x
0
)
contains
an open ball around
0
.
Then use the fact that
B
n
+
x
0
can be put inside a larger
ball around zero.
(iii) Show that for every point
u
in
B
r
(0)
there exists a point
x
in
B
2
(0)
such
that
Mx
=
u.
Hint: Using recursion and (ii), find the vectors
x
j
such that
u

∑
m
j
=1
Mx
j
<
r/
2
m
and
k
x
j
k
<
1
/
2
m

1
.
Note that
x
j
is an absolutely summable sequence.
(iv) Conclude that there is a
d >
0
so that
MB
1
contains
B
d
(0)
. This is the
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 Fall '09
 R
 Math, Topology, Metric space, Topological space, Banach space

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