# How can we define L-Infinity Norm for the multi-variable function?

How can we define L-Infinity Norm for the multi-variable function?

For $$\|f(t,x_1,\cdots,x_n)\|_\infty = ?$$ where $f \in C^\infty ([0,\infty) \times \mathbb R^n)$.

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Would you not define it basically the same way you define regular old $L^{\infty}(\mathbb{R})$ (the essential supremum)? –  tomcuchta Apr 21 '12 at 8:21
What happens if $f$ is not bounded? –  Davide Giraudo Apr 21 '12 at 9:18

Depends on the context. Sometimes you want $L^\infty_{x,t}$ which is the infimum of all numbers $M$ such that $|f(t,x)|\le M$ for a.e. $(x,t)$. In other situation you think of $t$ as evolving in time, and work with the norm $L^\infty_x$ obtained by fixing a value of $t$ and taking the infimum of all numbers $M$ such that $|f(t,x)|\le M$ for a.e. $x$.