functional analysis Haim brezis $4\mathrm{.}5$ this all the questions there is not any thing $1.$ Prove that L$^1(\backslash$Omega $)\backslash$cap L$^\backslash$infty $(\backslash$Omega $)$ is a dense subset of L$^$p$(\backslash$Omega $).2.$ Prove that the set $\{$f in L$^$p$(\backslash$Omega $)\backslash$cap L$^$q$(\backslash$Omega $)$ ;f$\_$q$<=1\}$ is closed in L$^$p$(\backslash$Omega $).3.$ Let $($f$\_$n$)$ be a sequence in L$^$p$(\backslash$Omega $)\backslash$cap L$^$q$(\backslash$Omega $)$ and let f in L$^$p$(\backslash$Omega $).$ Assume that f$\_$n$->$ f in L$^$p$(\backslash$Omega $)$ and f$\_$n$\_$q$<=$ C $.$ Prove that f in L$^$r$(\backslash$Omega $)$ and that f$\_$n$->$ f in L$^$r$(\backslash$Omega $)$ for every r between p and q$,$ r $!=$ q$.$