TITLE: How to show that the vector subspaces of $\mathbb{R}^{n}$ are closed in $\mathbb{R}^{n}$?
QUESTION [6 upvotes]: The vector subspaces of $\mathbb{R}^{n}$ are closed in $\mathbb{R}^{n}$.
How to show this?

REPLY [4 votes]: Any subspace $E\mathbb \subset R^n$ is very canonically the common zero  set of the family of all (continuous !) linear forms $\sum a_ix_i$ vanishing on $E$. [In the symbolism of annihilator sets : $E=E^{\circ\circ}$]