Problem: Let $X,Y$ be Banach spaces, $T:X o Y$ be a linear map. $T$ is said to be bounded, if $exists M>0$, such that $forall xin X, sen{Tx}leq Msen{x}$. Show that $T$ is bounded iff (if and only if) for any bounded subset $Bsubset X$, $T(B)$ is a bounded subset of $Y$.
Proof: $ a$: Let $B$ be a bounded subset of $X$. Then there exists a $R>0$, such that $sen{x}leq R, forall xin B$. Hence, $$ex sen{Tx}leq Msen{x}leq MR, forall xin B. eex$$ This verifies that $T(B)=sed{Tx;xin B}$ is a bounded subset of $Y$.
$la$: By the assumption, for $B=sed{xin X;sen{x}leq 1}$, $T(B)=sed{Tx; xin B}$ is a bounded subset of $Y$; that is, $$ex exists R>0,st sen{Tx}leq R, forall xin B. eex$$ Consequently, $$ex xin X af{x}{sen{x}}in B a sen{Tf{x}{sen{x}}} leq R a sen{Tx}leq Rsen{x}. eex$$