Topology and Geometry for Theoretical Physics & Functional Analysis and Spectral Theory.
DT234/DT238 FUNCTIONAL ANALYSIS
Homework Sheet
(1) Let U be an open set in a metric space (X, d). Let F = {1:1, . . .mm} be a finite
subset of U. Show that the set U F :2: E U,a: 9? F} is also open.
(2) (a) Let A and B be sets in the metric space (X, d). Show that the closure
AnBcAnB r
(b) Let W be subspace of a normed space V. Show that its closure W is again a
vector subspace of V.
(3) (a) Let (X, d) be a metric space and T : X -> X. Suppose T satisfies
d(T($),T(y)) < d($7 y)
when a: 75 y and has a fixed point. Show that the fixed point is unique.
(b) Let X 2 R with d(a:, y) :2 lac – yl, and let T(:13) 2 /$2 + 1. Show that
d($1,$2), V 171,1?2 E X,
but T does not have a fixed point. Does this contradict Banach’s Fixed Point
Theorem?
(4) Let (V1, – and (V5, ~ be normed spaces, and the product space V 2 V1 x V};
be endowed with the norm
(331, 332)” = max(ll$lll1a H552H2)-
Show that if V1 and V2 are Banach spaces then (V, is a a Banach space.
(5) On R” consider the norms = maxi-=1…” and “snug = (2le Show
that these two norms are equivalent.
2 DT234/DT238, HOMEWORK SHEET
(6) (a) Show that in a normed space (V, – the closed unit ball
31(0) = {w 6 VI llwll S 1}
is convex.
(b) Show that
f($1,fl72)=(/l331l+ V l-T2l)
does not define a norm on R2.(Hint: Sketch f = 1.)
(7) Let (V, – be a normed vector space. Prove that a linear functional f : V -> R
is continuous if and only if its kernel K er( f ) = {3: E V : f = O} is closed in V.
(8) Prove that the dual space of co is l1.
(9) Let the operator T be defined on the vector space V of all sequences by
$1 $2 $11,
T($1,$2,….’En,…) 2
Show that T is a linear operator.
(a) If T : ll -> [1, determine whether it is continuous. If it is7 find its norm.
(b) If T : loo -> l2, determine whether it is continuous. If it is, find its norm.
(10) Let coo be the normed space of sequences of real numbers with only finitely many
non-zero terms, and norm = supneN Let T : coo -> coo be defined by
a: x sun
T(:131,:l;2,…:vn,…)= ($1,32,33,…-;,…).
(a) Show that T is linear and bounded.
(b) Show that T“1 exists but is not bounded. Does this contradict Banach’s
Inverse Theorem?
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