Notation in analysis Could someone please explain the following notation
$$||f_j||_{\infty}$$
Its used in convergence theorems but I dont understand the double lines or the infinity symbol. 
 A: This could be the sup norm (uniform norm):
$$
\|f\|_\infty = \sup_x |f(x)|
$$
A: The  double bar notation $\| \cdot\|$ is sometimes used to denote some kinds of norms in mathematics. The question was already dealt with in What does double vertical-line means in linear algebra? A standard class of norms are the $\ell_p$ norms, with $1\le p < \infty$, in finite (or countably infinite) dimension spaces: 
$$ \| f\|_p = \left(\sum_i|f_i|^p \right)^{\frac{1}{p}}\,.$$
When $p\to \infty$, $\| f\|_p \to \| f\|_\infty = \sup_i |f_i| $.
Otherwise, $\left(\int|f(t)|^p  dt\right)^{\frac{1}{p}}$ denotes the  $L_p$ norm, and for $p = \infty$, it characterizes  essentially bounded functions, i.e. bounded up to a set of measure zero, as given in @Bungo comment.
The $_\infty$ notation is thus consistent with classical norms.
On the historical side: John Wallis is credited  with introducing the "lemniscatus" (from latin, and greek, decorated with ribbons) for the  infinity symbol with its mathematical meaning in 1655. The double-bar notation was introduced to denote matrices by Cayley (1843), but was apparently used by Erhard Schmidt in 1907 to introduce a norm notation.
