# The difficulties of number and the number of difficulties

This morning in a well known DIY store I heard two members of staff discussing the metre-cubed measure of volume. One said ‘what is a metre cubed?’ and the conversation then went:

‘Do you know what a square meter is?’

‘Yes’

‘Well that’s, like, with 2 and a metre-cubed is, like, with 3.’

‘Oh I see’

If this is a hard concept then, in the words of a well known fictional army reservist, ‘we’re doomed, doomed I say.’

# How short is your title?

What is the shortest journal paper title that you have come across? For me it is H = W, by Norman G Meyers and James Serrin (Proc Nat Acam Sci USA, 1964 – see http://www.pnas.org/cgi/reprintframed/51/6/1055).

The paper proves that $W^m_p(\Omega) = H^m_p(\Omega)$ where the first (Banach) space contains functions (distributions) whose norm,

$\Vert v\Vert_{W^m_p(\Omega)} := \left( \sum_{\vert\alpha\vert\le m}\Vert D^\alpha v\Vert_{L_p(\Omega)}^p \right)^{1/p}$

(with the usual $\mathrm{max}$ and $\mathrm{ess\, sup}$ adjustments when $p=\infty$), is finite while the second is defined as the closure of $C^\infty(\Omega) \cap W^m_p(\Omega)$ with respect to that norm. The result is true if $\Omega\subset\mathbb{R}^n$ is open but need not be if any part of the boundary is included.

By the way, not only is it a very short title, the paper itself is less than two sides.

The point of this entry to test LaTeX a bit more in this forum. I used Chrome to do this. IE seemed not to like \cap, and the LaTeX preview seems to get the math font sizes wrong…