Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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) $.

share|cite|improve this question
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. $x$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.