Vtyjkyuk

How to show

(i) the open unit ball in $(C[0,1],d_infty)$ is not open in $(C[0,1],d_1)$

(ii) the open unit ball in $(C[0,1],d_1)$ is open in $(C[0,1],d_infty)$ ?,

where $d_1(f,g)=int_0^1 vert f(x)-g(x) vert dx$ and $d_infty (f,g)=textsup_x; vert f(x)-g(x) vert$

I know the open unit ball in $(C[0,1],d_infty)$ is "the set of all functions which are less than the unit distance from the zero function" and the figure look like this:(outline of the graph)

Can I have a hint?

For 1) define $f_n (x) =nx$ for $0leq x leq frac 1 n$ and $0$ for other values of $x$. Note that the zero function $f$ is in the open unit ball for $d_infty$. Now $int_0^1|f_n-f|=frac 1 2n to 0$ and $|f_n|_infty=1$ for all $n$. Hence there is no $r>0$ such that $d_1(f,g) <r$ implies $g$ is in the open un it ball for $d_infty$. This proves 1).

For 2) suppose $int_0^1 |f(x)| , dx <1$. Let $r$ be a number between $int_0^1 |f(x)| , dx$ and $1$. Then $|f-g|_infty <1-r$ implies $int_0^1 |g(x)| , dx<int_0^1 |f(x)| , dx +1-r <1$. This prove that the ball of radius $1-r$ around $f$ is contained in the open unit ball so $f$ is an interior point.

Writing it out explicitly tends to make things clearer. Since the metrics involved are norms, let me write $|cdot|_p$ for the $p$ norm, and let $B_p$ be the open unit ball around 0 in the $p$ norm. For $(i)$, you want to find a function $f$ in $B_infty$ and a function $g$ with small 1-norm so that $f+gnotin B_infty$. Do you know of any positive functions that are huge in supremum norm but have small integral? You can take $f=0$.

For $(ii)$, you should again just write everything out. This time, you need to show that if you have a function $f$ in $B_1$, then there is a radius $r$ that can depend on $f$, such that for any function $g$ with $|g|_infty < r$, that $|f+g|_1 < 1$. Try to finish the proof.