# Unbounded linear operator

• March 30th 2011, 12:23 AM
raed
Unbounded linear operator
Dear Colleagues,

Let $X$ and $Y\neq \{0\}$ be normed spaces, where $dimX=\infty$. Show that there is at least one unbounded linear operator $T:X\longrightarrow Y$.
Let $(e_i)_{i\in I}$ a Hamel basis for $X$. Let $\left\{ f_n\right\}$ a sequence of linearly independent vectors of that basis. Let $v\neq 0\in Y.$ Put $T(f_n) = n\lVert f_n\rVert v$ and $T(e_i) =0$ if $e_i$ is not one of the $f_n$. You can show that $T$ is linear and not bounded.