# existance of the interpolation space

Let $X\subset L_1+L_2$ and let $Y$ be interpolation space between $L_2$ and $L_{\infty}.$ Given $U:X\longrightarrow Y$. My question is the following:

Is there exists space $Z\subset Y$, such that $U:X\longrightarrow Z$ and $Z$ is an interpolation space between $L_2$ and $L_{\infty}$?

Thank you.

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Could you clarify what properties $Z$ should have? Currently $Z=Y$ seems to satisfy the question, but this is surely not what you are looking for. –  Tom Cooney May 4 '12 at 16:32
Right now I am looking for any space $Z \subset Y$ the interpolation space between $L_2$ and $L_{\infty}$. Any such space with any properties would be good for me. –  David May 4 '12 at 17:26