Topology and Geometry for Theoretical Physics & Functional Analysis and Spectral Theory.

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? PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET AN AMAZING DISCOUNT :)

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