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There are three questions in total. IF AND ONLY IF question needs to prove by both side. Problem 1 contain 2 small questions, both are if and only if question. Problem 2 also needs to prove both side.
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[10] Problem 2. (cf. P8.5) Consider the normed vector space (10, 11• ||.o). Prove that
cl(coo) = Co.
See Lecture 8 and its Problem Section in your notes for relevant definitions.
cl(coo) sco:
vector
(10) Problem 3. Let (X, 11 · 1l) be a normed vector space. Let A be a nonempty open subset of
X and let B be a nonempty subset of X. Prove that the set
A+ B = {x E X: Ja € A, 36 E B = x = a + b}
is an open subset of X.
Problem P8.4 Let (X, 11• ID) be a normed vector space, and let W be a linear subspace
of X. Prove that cl(W) also is a linear subspace of X.
Note: In Problem P8.4, the linear space cl(W) may be strictly larger than W itself. An
example like this is shown in the next problem, where we consider the normed vector space
(20, 11•110o) introduced in Lecture 2. Recall (from Example 2.13) that
Coo :=
{x = ( x2(1), 2(2),..., z(K),...) € 40|3 ko such that 2(k) = 0 for k > ko}.
Let us also consider the linear subspace co Clao defined by
Co :=
{r = (3:1), 2121..........) € 4* | Lim z(K) =
lim x0).
Problem P8.5 Consider the normed vector space (8°, 11 ·||0o), and let the linear sub-
spaces Coo and c, of l® be as defined above. Prove that cl(coo) = Co.
[20] Problem 1. For i E N, let eį = (0,...,0,1,0,...) be the vector with 1 at the i-th place
and zeroes elsewhere. Let (1;)., be a sequence of real numbers and let f: Coo + R be a the
linear map uniquely determined by f(ei) = ; for all i E N.. 1987
(1) Prove that f is continuous with respect to || llo. (on coo) if and only if the sum
Li-l l; is absolutely convergent.
(f is continuous Doil is convergent):
(2) Prove that f is continuous with respect to || . ||ı if and only if the sequence (an)-1
is bounded.
(f is continuous (An) is bounded): Let Ines, VicN.
7
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