# Approximation of functions in $L^p(R^d;L^\infty)$

Assume that the function $$f(x,y)\in L^p(R^d;L^\infty(B_R))$$ with $$1, where $$B_R:=\{y\in R^d: |y|\le R\}$$. Can we find a class of functions $$f_n\in C_b^2(R^d;L^\infty(B_R))$$ such that $$\big\|\sup_{|y|\le R}|f_n(\cdot,y)-f(\cdot,y)|\big\|_{L^p}\to 0\quad as\quad n\to\infty ??$$

• Have you tried to approximate $f$ with a linear combination of simple function first, and then to approximate each simple function with a smooth one? – Skeeve Mar 25 '19 at 14:00
• I tried to approximate $f$ by functions of the form $f_n(x,y)=\sum_{i=1}^ng_i(x)h_i(y)$. But is it true that we can get$$\|\sup_{|y|\le R}|f_n(\cdot,y)-f(\cdot,y)|\|_{L^p}\to 0???.$$ – Wenguang Zhao Mar 25 '19 at 15:36
• I think this should be true if the condition $f\in L^p(R^d, X)$ is understood in the sense that $f$ is Bochner integrable with exponent $p$. – Skeeve Mar 25 '19 at 16:24
• The conclusion you mentioned seems to need the condition that $X$ is a reflective Banach space, is it right? – Wenguang Zhao Mar 25 '19 at 16:55
• I don't think reflexivity is needed here, see e.g. Proposition 2.1.10 in the book Nonlinear Analysis by Leszek Gasinski, Nikolaos S. Papageorgiou – Skeeve Mar 25 '19 at 17:04