I am trying to find a sequence of functions $\{f_j\}$ and a function $f$ such that $f_j\to f$ strongly in $L^2(\mathbb R)$ but $f_j$ does not converge to $f$ pointwise. The definition of strong convergence I am given is:
$f_j\to f$ strongly in $L^2$ if there exists $f\in L^2\mathbb R$ such that $$||f_j-f||_{L^2(\mathbb R)}\to0$$
Now I am not interested at all in a full solution to this, I already have it and am choosing not to look at it. Rather, I'd like to know how one would approach a problem like this. Should I start by defining an inner product to find the norm, so as to at least gain intuition for what the norm represents? Should I just start trying common functions and 'tweak' as appropriate? Should I instead write out the epsilon-delta definition of a limit and try to expand from there using norm properties? Should I just try all of those? How does one even know where to start?
Usually, I can develop some intuition for the problem but inner products in a space need not be unique so the abstractness makes it much harder for me I find.
On $([0,1],\mathscr{B}([0,1]),\lambda)$ where $\lambda$ is Lebesgue's measure consider the sequence
$X_1=\mathbb{1}_{[0,1]}$
$X_1=\mathbb{1}_{[0,1/2]}$, $X_3=\mathbb{}_{[1/2,1]}$,
$X_4=\mathbb{1}_{[1,1/4]}$, $X_5=\mathbb{1}_{[1/4,2/4]}$,$X_6=\mathbb{1}_{[2/4,3/4]}$, $X_7=\mathbb{1}_{[3/4,1]}$.
... see the pattern?
Once can see that $\|X_n\|_2\rightarrow0$ (the subintervals get smaller and smaller) but $0=\liminf_nX_n(t)<\limsup_nX_n(t)=1$ for all $t$ (like the typewriter got from one en of a line to the other)