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If $f_0\ne0$ or if $f_0=0$ and the $f_1,\dotsc,f_n$ are not linearly independent, then the answer is trivial: In this case there is another functional, say $f_1$, which is in the span of the remaining …

Diestel & Uhl can only mean the first interpretation you gave for two reasons: For the second interpretation, the term “off a fixed set of measure zero” makes no sense. Even in case $X=\mathbb R$ (wh …

Only now I realize the condition that $\zeta$ is nonnegative. (Was it really there in the first formulation of the question?) With this condition, it is possible to get the required a-priori bound req …

It's many years ago that I read it, but I think that some of the most general interpolation type results for Orlicz spaces were contained in O’Neil, Richard, Integral transforms and tensor products on …

The definition makes no sense due to the mixing up of "relatively" (weakly) compact and (not relatively) compact. I guess that what you mean is: $\psi$ is strongly-weakly proper on closed balls (that …

What is needed is an a-priori $L_\infty$ bound for the solution $v_k$. If you know such an a-priori bound, you can modify $\zeta$ outside of this bound, and you can assume without of generality that $ …

Maybe the simplest classical example is a weakly singular kernel $$K(x,y) = |x-y|^{-\lambda}$$ with some fixed $\lambda\in(0,1)$. In this example $\int_{\mathbb R^2}K(x,y)^qdx=\infty$ for every $q>0$ …