I want to find a sequence (f_k)of continuous bounded functions on [0,1]
converging pointwise to a continuous limit, but such that
the sup-norm of f_k tends to infinity when k tends to infinity...
is there any?
I want to find a sequence (f_k)of continuous bounded functions on [0,1]
converging pointwise to a continuous limit, but such that
the sup-norm of f_k tends to infinity when k tends to infinity...
is there any?
Nope, i don't think there are any. Since if $\displaystyle f_k \rightarrow f $ pointwise then
$\displaystyle \lVert f_k \lVert_{sup} \rightarrow \lVert f \lVert_{sup} $ which would mean f is not bounded on [0, 1] (since $\displaystyle \lVert f_k \lVert_{sup} \rightarrow \infty $ ) thus it must not continuous on [0, 1].
What you need to do is to ensure that the function $\displaystyle f_k$ has a narrow spike somewhere. For example, define $\displaystyle f_k(x) = k^2x$ in the interval $\displaystyle 0\leqslant x<1/k$, $\displaystyle f_k(x) = \tfrac2k - k^2x$ in the interval $\displaystyle 1/k\leqslant x<2/k$, and $\displaystyle f_k(x) = 0$ in the remainder of the unit interval. Then $\displaystyle f_k(x)\to0$ as $\displaystyle k\to\infty$, for each (fixed) point x in the unit interval.
That is not true, because pointwise convergence does not imply convergence in the sup norm.