Note that $\dim E$, if it exists, is ......
There exists a function $f$ and a constant $c$ such that ...... [Or: There exist a function $f$ and a constant $c$]
The limit $\lim_{x\to 0}f(x)$ exists. [Not: “There exists a limit $\lim_{x\to 0}f(x)$.”]
What relations exist between $A$ and $B$?
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