Around these parts, the aphorism "A gentleman never chooses a basis," has become popular.

Is there a gentlemanly way to prove that the natural map from V to V^** is surjective if V is finite-dimensional?

As in life, the exact standards for gentlemanliness are a bit vague. Some arguments seem to be implicitly picking basis. I'm hoping there's an argument which is unambiguously gentlemanly.