prove these theorems in detail please.

THEOREM : Let B ( x, Y) be the family of all bounded linear maps from normed space X to normed space y B ( X , Y ) = {f : x - y , f is bounded and linear ) For an arbitrary fE B ( X, y ), the following conditions are equivalent . ( a) litll = Sup 1/toll *#0 11X11 ( b ) 1Itll = int { x : 11 f (2 )1/ = Kl1x/1 ] , K=0 ] ( @ ) All = Sup 11x71 = 1 THEOREM : Let x be a normed space and let do to be an arbitrary element of X. Then there exist a bounded linear function f on X. Such that If( x21 1 = 1 and f (2) = 11 Xall