'Smaller than infinity' notation

Question

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?

Answer 1

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.

Answer 2

In some contexts one adjoins two objects to the real line called $\infty$ and $-\infty$ so that one is working in the set $\mathbb R\cup\{\infty\}\cup\{-\infty\}$, and the less-than relation among the members of this set is just what you would expect. In that case the only things less than $\infty$ are real (in particular, finite) numbers and $-\infty$.

There are many different things in mathematics that are either called "infinity" or are thought of as in some sense infinite quantities. We've had other questions posted here about that, and some good answers. You need to bear in mind which sort of "infinity" is the right one in the context you're working in. Something similar in spirit to $\pm\infty$ as contemplated here is a single $\infty$ at both ends of the line, so the line plus that object becomes topologically a circle. Then one can say that $\tan\frac\pi2=\infty$ and $\tan : \mathbb R\to\mathbb R\cup\{\infty\}$ is continuous at every point in the domain $\mathbb R$. The infinities that are cardinalities of infinite sets are something different. The "points at infinity" in projective geometry are yet another concept. The "infinity" that is the value of Dirac's delta function at $0$ is yet another kind of infinity (in particular, it makes sense, for example, to multiply it by $3.2$ and get a different "infinity". The infinite numbers of Robinson's nonstaandard analysis are quite different things from all of the above. And there are other examples.