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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