How is continuity different between a sequence of points in $\mathbb{R}^n$ and a sequence of functions in a function space? For any continuous function $f(x)$ we have that
$$
(*) \quad \quad \lim_{x\to x_0} f(x) =  f(\lim_{x\to x_0}x) = f(x_0).
$$
Because of this, I always believed that by the continuity of the norm function, for a sequence of functions $f_n$ converging to some function $f$, that we could say
$$
(**) \quad \quad \lim_{n\to \infty} ||f_n||_p =  || \lim_{n\to \infty} f_n||_p = ||f||_p.
$$
So is there a difference between $(*)$ and $(**)$? Are we not allowed to perform the switching of limits in $(**)$, even though the norm is continuous?
On a related note I also always thought we could do the following for a sequence of functions in a Hilbert space:
$$
\lim_{n\to \infty} \langle f_n, g \rangle = \langle   \lim_{n\to \infty} f_n, g \rangle = \langle f, g \rangle.
$$
Is this also not true?
 A: In the setting of functional analysis, one needs to understand the notation $$
\lim_{n\to\infty}f_n=f\tag{1}
$$ 
with contexts since there are different modes of convergence. Usually, 
(1) is understood as norm convergence in some Banach space. For other modes of convergence, one would write
$$
f_n\to f\quad\hbox{ [in some mode]}
$$
where [in some mode] could be "almost everywhere", "pointwise", "in measure", "weakly", etc. assuming one has extra appropriate structures. 
The statement ($**$) in your question should be interpreted precisely as

if $f_n\to f$ in norm $\|\cdot\|_p$, which by definition means $\|f_n-f\|_p\to 0$, then $\|f_n\|_p\to\|f\|_p$.

This is can be easily proved by the triangle inequality:
$$
|\|f_n\|_p-\|f\|_p|\leq \|f_n-f\|_p.
$$
Again, in
$$
\lim_{n\to \infty} \langle f_n, g \rangle = \langle   \lim_{n\to \infty} f_n, g \rangle = \langle f, g \rangle,\tag{2}
$$
the notation $\lim_{n\to \infty} f_n=f$ should be understood as convergence in the Hilbert space, namely, convergence in norm induced by the inner product. It is a simple exercise by Cauchy-Schwarz:
$$
|\langle f_n,g\rangle-\langle f,g\rangle|\leq \|f_n-f\|\cdot \|g\|.
$$
