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Math 175 Homework 1
Due Thursday, April 8
1. (a) Show that if a (real or complex) vector space
V
has an inﬁnite dimensional
subspace, then
V
is inﬁnite dimensional.
(b) Consider the normed linear space
C
([0
,
1])
of real valued continuous func
tions on
[0
,
1]
with its usual norm (i.e.
k
f
k
= max
[0
,
1]

f

) and let
S
=
span
±
1
,x,x
2
,...
²
.
Prove that
S
is not a closed subspace of
C
([0
,
1])
, and hence in particular that
S
is
inﬁnite dimensional (hence
C
([0
,
1])
is inﬁnite dimensional by (a)).
Notes: (a)
V
is ﬁnite dimensional if we can ﬁnd a ﬁnite collection of vectors
which span
V
.
V
is inﬁnite dimensional if it is not ﬁnite dimensional.
(b) Here (and subsequently)
span
{
A
}
(with
A
a nonempty subset of a linear
space
X
) will mean the set of all possible linear combinations of ﬁnite collections
of vectors taken from
A
. Thus,
span
{
A
}
=
n
∑
N
i
=1
λ
i
a
i
o
,
where
λ
i
are scalars
and
a
i
∈
A.
(This is what your text calls
lin
(
A
)
.)
2. Let
l
2
R
denote the set of all real sequences
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 Fall '09
 R
 Vector Space

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