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Thread: Unbounded linear operator

  1. #1
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    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.
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  2. #2
    Super Member girdav's Avatar
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    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.
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