Unbounded linear operator

**raed** · March 30th 2011, 12:23 AM

Dear Colleagues,

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

Regards,

Raed.

**girdav** · March 30th 2011, 10:41 AM

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.

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