Separation axioms for topological vector spaces

Prove that a linear topological space satisfies the third separation axiom $T_{3}$.

Prove that a linear topological space is separated (i.e., satisfies the first separation axiom $T_{1}$) if and only if the intersection of all neighborhoods of zero consists only of zero. In particular, every locally bounded linear topological space is separated.

Prove that a polynormed space $X$ is Hausdorff with respect to the topology generated by the system of its seminorms $\left\{ p_{\alpha }\right\}_{\alpha \in A}$ if and only if for every vector $x \in X, x \neq 0$, there exists a seminorm $p_{\alpha }, \alpha = \alpha (x)$, such that $p_{\alpha }(x) > 0$.

Prove that the spaces $s, l_{2}^{\infty }, C^{\infty }[0,1], \mathcal{A}(D), C(\mathbb {R})$ and $C_{\mathrm{e}}[0,1]$ are Hausdorff.

Suppose that $\mathscr {X}$ is a TVS but do not assume that $\mathscr {X}$ is Hausdorff.

Let $V$ be a neighborhood of 0 in a topological vector space $X$. Prove that there is a real continuous function $f$ on $X$ such that $f(0) = 0$ and $f(x) = 1$ outside $V$. (Thus $X$ is a completely regular topological space). Suggestion: Let $V_{n}$ be balanced neighborhoods of 0 such that $V_{1}+V_{1} \subset V$ and $V_{n+1}+V_{n+1} \subsetneq V_{n}$. Construct $f$ as in the proof of Theorem 1.24. Show that $f$ is continuous at 0 and that