I'm studying linear algebra and analysis. I'm not a English user so that some sentences are confusing to me:
"There exists an element in $V$ (vector space) denoted by $0$ such that $x+0=x$ for each $x$ in $V$"
What does it mean that "There exists an element in $V$"?
Does it mean that there exists only one element in $V$? Because "an" is one, right?
A similar problem is in analysis too.
"Every infinite subset of $K$ has a limit point in $K$"
Does it mean that every infinite subset of $K$ has only one limit point in $K$?
Best Answer
The first sentence is a bit confusing to non-English speakers. On the one hand, the word "an" is an indefinite article indicating there is at least one, but possibly more. On the other hand, when we refer to "an element $\ldots$ denoted by $0$," we're implying that there is only one such element (because we're naming it).
This is actually foreshadowing. We are saying that there is at least one such element but it is very easy to prove (by considering $0_1+0_2$ if there were two such elements) that the element must be unique.
The second sentence unambiguously says merely that there is at least one limit point.