What do L¹(ℝ) and the dot notation in ||u(·,t)||_L∞ mean?

[Arkuski]

I have a homework problem discussing the heat equation, and I have two questions about notation that I could not figure out through google searching:

$L^1(ℝ)$ as in a function $g\in C^1(ℝ)\cap L^1(ℝ)$

$||u(\bullet ,t)||_{L^∞}$

The bullet in the second expression is actually a single dot at that height, but I could not find the latex symbol for it. I initially thought it was a limit as $x\rightarrow\infty$ but that is incorrect.

 

[Dick]

$L^1$ is the set of all functions f such that |f| has a finite integral. $L^\infty$ is the max norm. Since the dot is in the first position it's the max of |u(x,t)| over all x for a fixed t if u is continuous. If u is not continuous, it's the max possibly excluding a set of measure 0. If you are dealing with the heat equation you probably don't have to worry about the 'measure 0' thing.

 

[brmath]

latex note

for what it is worth, the latex notation for the dot as in dot product is \cdot.

 

[Arkuski]

Thanks for the help guys!