# FA2

(i) Let $X=C[0,1]\,$ with maximum metric $d(f,g)=\max\{|f(t)-g(t)|:t\in [0,1]\}\,$. Prove that $C[0,1]\,$ is separable.

(ii) Use (i) to prove that $X=C[0,1]\,$ is also separable with respect to the metric $\rho (f,g)=\left(\int _{0}^{1}|f(t)-g(t)|^{p}dt\right)^{{1/p}}\,$.

## (i)

Let $X=C[0,1]\,$ have the maximum metric $d(f,g)=\max\{|f(t)-g(t)|:t\in [0,1]\}\,$. Let $Y\,$ be the set of all polynomials with rational coefficients on $[0,1]\,$. Let $Z\,$ be the set of all real polynomials on $[0,1]\,$. Since $Z\,$ is dense in $X\,$, ${\bar {Z}}=X\,$.

$Y\,$ is countable. Proof: Let $Y_{n}\,$ be the set of all polynomials of degree $n\,$ with rational coefficients. Then $|P_{n}|={\mathbb {Q}}\times {\mathbb {Q}}\times ...\times {\mathbb {Q}}\,$, where there are $(n+1)\,$ ${\mathbb {Q}}\,$'s. Now define $Y=\bigcup _{{n=1}}^{\infty }P_{n}\,$. $Y\,$ is countable because it is a countable union of countable sets.

Let $f\in Z\,$ such that $f(t)=\sum _{{j=0}}^{n}a_{j}t^{j},a_{j}\in {\mathbb {R}}\,$. Let $\epsilon >0\,$. Since ${\bar {{\mathbb {Q}}}}={\mathbb {R}}\,$, there exists $q_{0},q_{1},q_{2},...,q_{n}\in {\mathbb {Q}}\,$ such that $|q_{i}-a_{i}|<{\frac {\epsilon }{n+1}}\,$ for $i=0,...,n\,$. Let $g(t)=\sum _{{i=0}}^{n}q_{i}t^{i}\in Y\,$.

Notice $d(g,f)=\max _{{t\in [0,1]}}|g(t)-f(t)|\,$

$=\max _{{t\in [0,1]}}|\sum _{{i=0}}^{n}(q_{i}-a_{i})t^{i}|\,$

$\leq \max _{{t\in [0,1]}}\sum _{{i=0}}^{n}|q_{i}-a_{i}|t^{i}\,$ by the triangle inequality.

$=\sum _{{i=0}}^{n}|q_{i}-a_{i}|\,$

$\leq {\frac {\epsilon }{n+1}}(n+1)=\epsilon \,$.

$g\in B(f,\epsilon )\,$, making $f\in {\bar {Y}},Z\subseteq {\bar {Y}}\,$. We have $X={\bar {Z}}\subseteq {\bar {{\bar {Y}}}}={\bar {Y}}\,$. Furthermore, ${\bar {Y}}\subseteq X\,$, so ${\bar {Y}}=X\,$. Since $Y\,$ is countable, $X\,$ is separable.

## (ii)

Let $X,Y,\,$ and $Z\,$ be defined as above with the metric $d(f,g)=\left(\int _{0}^{1}|f(t)-g(t)|^{p}dt\right)^{{1/p}}\,$. We still have ${\bar {Z}}=X\,$and $Y\,$ is countable. Let $f\in Z\,$ such that $f(t)=\sum _{{i=0}}^{n}a_{i}t^{i},a_{i}\in {\mathbb {R}}\,$. Let $\epsilon >0\,$. Since ${\bar {{\mathbb {Q}}}}\subseteq {\mathbb {R}}\,$, there exists $q_{0},q_{1},...,q_{n}\,$ such that $|q_{i}-a_{i}|<{\frac {\epsilon }{n+1}}\,$. Notice $g(t)=\sum _{{i=0}}^{n}q_{i}t^{i}\in Y\,$. From (i), we have $d(g,f)<\epsilon \,$. Now notice, $\rho (f,g)=\left(\int _{0}^{1}|f(t)-g(t)|^{p}dt\right)^{{1/p}}\,$. $\leq \left(\max _{{t\in [0,1]}}|f(t)-g(t)|^{p}\right)^{{1/p}}\,$ $=\max _{{t\in [0,1]}}|f(t)-g(t)|\,$ $=d(f,g)<\epsilon \,$

$g\in B_{\rho }(f,\epsilon )\,$, and by the same argument as in (i), $X\,$ is separable.