What is the meaning of expressions of the type $f(\cdot)$ (function (dot))?

Question

Simple question, fully expressed in the Title line. Is the dot within the parenthesis intended to mean, "any possible function"?

Answer 1

Usually we use it to avoid writing more letters $x$, $y$, etc. One example I see a lot: let $B: V \times V \to W$ be a bilinear form, and fix $y \in V$. When we write $B(\cdot, y)$, we mean the map $$V \ni x \mapsto B(x,y) \in W,$$ so we don't write this extra $x$ if we don't need to. If we're going to write $f(\cdot)$ just like this, as in the title question, then there isn't much advantage - just talk about the function $f$ and be done with it. The advantage I see is where you want to simplify the writing of some function that uses another one "in the background", like the example with the bilinear form I gave above.