$F:\Big(C[0,1],||.||_2\Big)\rightarrow \Big(C[0,1],||.||_3\Big)$
$x\rightarrow F(x)(t)=\int^t_0x(s)ds,\quad\quad0 \le t\le 1 $
Show that F is continuous.
F is linear. for n=0,1,2.. $x_n(t)=t^n,0 \le t\le 1 $
I found $||F(x_n)||_3 =\frac{1}{(n+1)^3\sqrt[3]{3n+4}}$
and $||x_n||_2=\frac {1}{\sqrt{2n+1}}$
so $sup\frac{||F(x_n||_3}{||x_n||_2}=0$ when $n\to \infty$
May I say it is continuous because it is linear and $sup\frac{||F(x_n||_3}{||x_n||_2}\le +\infty$ ?