Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.

Then by intermediate space construction, usually the $K$-method or $J$-method, reference to this book, Chapter 7, that we could construct a intermediate space $X_s$ such that $$ X_0\hookrightarrow X_s\hookrightarrow X_1 $$ with $K$-norm or $J$-norm on $X_s$. (Both of those norms have long definitions, I don't want to copy everything here but you can definitely find them in the reference book I mentioned)

My question: If, in addition, we assume that $X_0\hookrightarrow\hookrightarrow X_1$, i.e., $X_0$ is COMPACT embedded in $X_1$, then can we construct intermediate space $X_s$ such that $X_s\hookrightarrow\hookrightarrow X_1$ as well?

If yes, please direct me to a reference.

Thank you!


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