Polarity is symmetric Let X$\subseteq \mathbb{R}^n$ be a closed, convex set that contains the origin: $\textbf{0}_n \in$X. The polar of X is $X_{po}:=\\{y\in \mathbb{R}^n : x^T y\leq 1,\forall x \in X\\}$ I'd like to convince myself that $(X_{po})_{po}$=X, using 1st principles (for instance, no functional analysis arguments). I've already shown X$\subseteq (X_{po})_{po}$, it's just the opposite inclusion that I'm stuck on.

Suppose that there exists some $\mathbf p\in (X_{\text{po}})_{\text{po}}$ such that $\mathbf p\
otin X$. Since $X$ is closed and convex, a separation argument reveals the existence of some $\mathbf y\in\mathbb R^n\setminus\\{0\\}$ and $c\in\mathbb R$ such that $\mathbf p^{\top}\mathbf y>c> \mathbf x^{\top}\mathbf y$ for each $\mathbf x\in X$. Since $\mathbf 0\in X$, one has $c>\mathbf 0^{\top}\mathbf y=0$; also, upon rescaling $\mathbf y$ if necessary, one can assume without loss of generality that $c=1$, so that $\mathbf x^{\top}\mathbf y<1$ for each $\mathbf x\in X$. This implies that $\mathbf y\in X_{\text{po}}$. Since $\mathbf p\in (X_{\text{po}})_{\text{po}}$, it follows that $\mathbf p^{\top}\mathbf y\leq 1$, which contradicts $\mathbf p^{\top}\mathbf y>c=1$.