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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.

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  • $\begingroup$ $||f||_\infty$ 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 $1$) $\endgroup$ – Nick Alger Dec 18 '15 at 20:42
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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.

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This could be the sup norm (uniform norm): $$ \|f\|_\infty = \sup_x |f(x)| $$

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

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