Thread: Unbounded linear operator

Dear Colleagues,

Could you please help me in solving the following problem:
Let $\displaystyle X$ and $\displaystyle Y\neq \{0\}$ be normed spaces, where $\displaystyle dimX=\infty$. Show that there is at least one unbounded linear operator $\displaystyle T:X\longrightarrow Y$.

Regards,

Raed.

Let $\displaystyle (e_i)_{i\in I}$ a Hamel basis for $\displaystyle X$. Let $\displaystyle \left\{ f_n\right\}$ a sequence of linearly independent vectors of that basis. Let $\displaystyle v\neq 0\in Y.$ Put $\displaystyle T(f_n) = n\lVert f_n\rVert v$ and $\displaystyle T(e_i) =0$ if $\displaystyle e_i$ is not one of the $\displaystyle f_n$. You can show that $\displaystyle T$ is linear and not bounded.