Dual pair

Remarks

Some authors call any triple satisfying condition 1 a dual pair, and call it separated when 2 and 3 hold. Example: $X = Y = \R^n$ with $\langle x, y \rangle = \sum_i x_i y_i$; if $x \ne 0$ then $\langle x, x \rangle > 0$, which gives 2, and 3 holds in the same way. Example: for a Hausdorff locally convex space $E$, the triple $(E, E', \langle x, f \rangle = f(x))$ is a dual pair; condition 2 follows from the Hahn–Banach theorem, and condition 3 is trivial. Non-example: $X = C([0, 1])$, $Y = \R$, $\langle f, t \rangle = t f(0)$ satisfies 1 and 3 but not 2, since the function $f(s) = s$ is nonzero and $\langle f, t \rangle = 0$ for all $t$.