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Let $Y=\{x\in l^p:x(2n)=0\}$, $1\leq p \leq \infty$. It can be proved that $Y$ is closed subspace of $l^p$. Define the quotient space $l^p/Y=\{x+Y:x\in l^p\}$. Then, by the fact that $Y$ closed, a norm on $l^p/Y$ can be defined by $||x+Y||=\inf\{\|x-y\|_p : y\in Y\}$. Can we find a surjective linear transformation $T:l^p/Y\rightarrow l^p$ such that $\|T(x+Y)\|_p=\|x+Y\|$?

My effort: For case $p=2$, $l^p$ is a Hilbert space, so there is an inner product that induce the norm, hence we can define orthogonal projection on $l^2$. By using this, we can verify $T(x+y)=x-P_Y(x)$ (where $P_Y$ is orthogobal projection on $l^2$) is a surjective linear transformation $T:l^2/Y\rightarrow l^2$ such that $\|T(x+Y)\|_2=\|x+Y\|$.

Please help me to solve the problem for $p\neq 2$.

Thanks a lot.

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1 Answer

Hint: $\|x + Y\| = \left( \sum_{j} |x_{2j}|^p \right)^{1/p}$.

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