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What are the easiest examples of a pairs of Banach spaces $X,Y$ such that

  • $X\subseteq Y$ ($X$ is a closed linear subspace of $Y$)
  • there is a bounded linear map $T\colon X\to Y$;
  • there is no bounded extension $\hat{T}\colon Y\to Y$ of $T$?

Needless to say, I am interested in the structure of the operator $T$ rather than in its existence.

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Consider $X = C^1(0,1)$ and $Y = C^0(0,1)$ and the map $T = \frac{d}{dx}: C^1(0,1) \to C^0(0,1)$. Every densely defined unbounded operator will provide an example.

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A little ambiguity: What if we asked $X$ be equipped with the norm induced by $Y$? $X$ would have to be non-complemented in $Y$, but it's the most I get... – Jose27 May 4 '12 at 1:42
Yes, $C^0(0,1)$ is not a Banach space under sup norm... Of course, I meant $X\subseteq Y$ and the norm is induced. – Rumburak May 4 '12 at 7:27
@Rumburak: Of course, it is your job to ask the question unambiguously... but I'll add an example later on. – t.b. May 4 '12 at 7:36

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