# 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.

• $||f||_\infty$ is the largest value that the function $f$ takes. Or, the smallest value larger than any value $f$ takes, in the case where the maximum value is never exactly achieved (e.g., there is a sequence of points $x$ with $f(x)=0.9$, $0.99$, $0.999$, etc, but never exactly $1$) – Nick Alger Dec 18 '15 at 20:42

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.
This could be the sup norm (uniform norm): $$\|f\|_\infty = \sup_x |f(x)|$$
• Or perhaps the $L^{\infty}$ norm, which ignores "outliers" on a set of measure zero: $\|f\|_{\infty} = \inf\{c \geq 0 \mid |f(x)| \leq c \text{ almost everywhere}\}$ – Bungo Dec 18 '15 at 20:46