FA2

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(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}}\,.

Contents

(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.


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