[Math] ‘Smaller than infinity’ notation

infinitynotation

I've been coming across some papers (written in the 1960s – 1970s) that use the following peculiar statement:

Let use denote by $H$ the space of all grid-functions $w_r$ for which:
$$ \sum_{\nu=1}^{\infty} |w_{\nu}|^2 h < \infty$$

Clearly, the author intends to say that the above sum should be finite. However, I find this notation rather unclear. Surely you can't just say that something is smaller than infinity? Even infinity is smaller than infinity, if you really wanted it to be, I thought the point is that inequalities make little sense when infinities are concerned.

Is the above considered a correct formal notation? Or is there a better way of expressing the same thing?

Best Answer

When dealing with a sum of nonnegative terms $\sum_{n=1}^\infty a_n$, there are only two things such a sum can do:

  1. it converges to a finite value $L$, in which case we say $\sum_{n=1}^\infty a_n = L$, or
  2. it diverges to $+\infty$, in which case we say $\sum_{n=1}^\infty a_n = \infty$.

In case (1.), if we don't care to mention (or don't know) the value $L$, we can write $\sum_{n=1}^\infty a_n < \infty$. This is perfectly standard notation, used in almost any text of analysis.

This also applies to integration of a nonnegative function with respect to a positive measure.

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