Question

Exer ut X Banach space BLX) = 3 TX X and linn and Bounded? | те а че ти чи є x хи зі ром свех », -и) в диаси асоvе liis now let T compart oprutor 5(x). { те всх), тoтp«3 ора!» У f(x) = 8 TE BEX) it has finite Rank} prev (5cx) , 1-1 ) , (w), 1-) are Banach algebra

Answer 1

112111 et & be a Banach space. Then B(x) = {T: X~x linear and bounded? and 11 Tll = sup 11 Tall HEX. we know that sup Itall a sup { "T?!! . Folex } . T1 = sup { TO) 6406x3 The usual operator worm makes B (x) into a normed vector space and is S, TEBCX) Shen 115T|| = sup {"TSC))} 111111 11211) I T(S01011 )S(1) Thail : Soto 2 mscoull < !! TUJISII This B(x) is a normed algebra. he For completeness, it & ll Tall Converges then so does t boruto ad Na ani so & Tn (2) enish since x is a Banach opgee it is clear that T(x) = 5T (21) debined Pointwise is in bact bounded and linear. ..B(x) is Banach algebra with the operator norm CITI & Sup Tall & Nex) as its norm, 1791/11