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This is true. See corollary of theorem 1.14 in W. Rudin Functional analysis. This property doesn't imply LCS have a countable separating family of seminorms.

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Yes, any locally convex space has a neighborhood basis around zero consisting of balanced convex sets (Wikipedia seems to take this as the definition but as far as I know including balanced" in the definition is non-standard, and in any case can be derived from the more general definition requiring the existence of a neighborhood basis around zero ...

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This isn't a full answer, just a simple observation that spectral theory gives a partial answer of sorts (as you may already know). See here or any introduction to Banach algebra/C*-algebra theory for supporting details. A character of a unital commutative Banach algebra $A$ is a unital algebra homomorphism $\phi: A \to \mathbb{C}$. The set $\Phi_A$ of all ...

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Lemma 1 If $X$ is first countable and $x_{n}\rightarrow 0$, then there exist positive scalars $\lambda_{n}\rightarrow\infty$ such that $\lambda_{n}x_{n}\rightarrow 0$. Proof: Let $\left\{U_{k}\right\}$ be a balanced local base for $0\in X$ such that $U_{k+1}\subset U_{k}$. Since $x_{n}\rightarrow 0$, there exists an increasing sequence $(N_{k})$ of ...

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Any completely regular (Hausdorff) topological space can be imbedded in $\mathbb{R}^J$ for some index set $J$, which is a topological vector space. Indeed, this is true if and only if a topological space is completely regular. Thus any Hausdorff topological space that is not completely regular cannot be imbedded in a topological vector space over the reals.

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first reduce to the case where $m=p.$ To do this rotate so that the image of the differential map is the first $p$ coordinates and then project onto the first $p$coordinates. If this map has rank $p$ nearby that is enough. If $m =p$ then it follows from the inverse function theorem. If $m>p,$ then rotate the domain to make the differential applied to ...

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