Is there a "geometric" intepretation of weak convergnce in terms of the graph of the functions or function space geometry? I think about weak and weak* convergence as pointwise convergence on the induced functional on the dual and prelude respectively.
Is there a way to think about this geometrically either in graph of the function or in the function space? Feel free to impose restrictions!
 A: 
Feel free to impose restrictions!

Ok, take for a simple case $f_n \in L^2[0;1]$ to be real valued functions (complex valued is more difficult to draw). It's easy to show that $f_n$ converges weakly to $0$ imply $\int_A f_n d\lambda\to 0$ for every measurablesubset $A\subset [0;1]$. Indeed $\int_A f_n d\lambda=\left\langle \chi_A,f_n\right\rangle_{L^2[0;1]}$ and the definition of weak convergence gives us the limit.
This means that $f_n$ is either close to $0$ (and in that case it strongly converges to $0$ around that point) or "oscillating" very quickly and evenly around $0$ (evenly in the sens that $\int_A f_n d\lambda \to 0$). One good example of that is given by Lebesgue-Riemann's lemma, $f_n:t\mapsto \sin(2\pi nt)$ weakly converges to $0$ but it doesn't converges strongly in any point so it must oscillate quickly.
It can be shown (but it's more difficult, you can do it as an exercise) that if $\int_A f_n d\lambda\to 0$ for every measurable set $A$ and if $\|f_n\|_2$ is bounded then $f_n$ weakly converges to $0$. One can obviously replace $0$ by $g\in L^2$ and $f_n\rightharpoonup g$ by simply considering $f_n-g$.  
