What is the difference between a "sharp bound" and a "tight bound"? Are the two adjectives synonyms in mathematical prose? Otherwise, when would you use one and when the other?
Very basic example just to give some context: the quantity $\|a\| \|b\|$ is a sharp/tight bound for $a\cdot b$. But this example might be insufficient, especially if there are nuances in meaning: I am interested in all usages of the two terms when dealing with inequalities / bounds / estimates. I do realize that there might be no exact mathematical definition of these terms.
My understanding is that a sharp bound is one that is achieved, which also implies that it cannot be improved (e.g., for an upper bound, no smaller value is an upper bound). A tight bound cannot be improved, but is not necessarily achieved.
For example, $x \leq 1$ is a sharp bound for $x\in [0,1]$, but is only a tight bound on $(0,1)$. On either interval, $x\leq 2$ is neither sharp nor tight, but is still a bound.