Let $X,Y$ be topological vector spaces and $T:X\rightarrow Y$ a linear transformation. Show that $T(X)$ is a bounded set in Y iff $T(X)\subset \cap _{V\in N} V$ where $N$ is a neighborhood base at $0$ of $Y$.
So if we assume $T(X)\subset \cap _{V\in N} V$ then it is easy to show that $T(X)$ is a bounded set in Y.
Now if we suppose $T(X)$ is a bounded set in Y, Then given any $V\in N$ there exists $t>0$ such that $tT(X)\subset V$. If $t\geq 1$, I am done because $V$ is balanced. But what if $t<1$? I am stuck now. Can somebody please help me?