# A Hausdorff topological vector space is locally convex if and only if $0$ has a neighborhood base of balanced, convex, absorbing space

The following is Theorem V.1 in Reed & Simon's book on functional analysis.

Let $$V$$ be a vector space with a Hausdorff topology in which addition and scalar multiplication are separately continuous. Then $$V$$ is a locally convex space if and only if $$0$$ has a neighborhood base of balanced, convex, absorbing sets.

In their proof they reference the following lemma about the Minkowski functional or gauge of an absorbing set $$C$$ with the additional property that if $$x \in C$$ and $$0 \leq t \leq 1$$, then $$tx \in C$$.

(a) If $$t \geq 0$$, then $$\rho(tx) = t \rho(x)$$ for the gauge of any set $$C$$.

(b) $$\rho$$ obeys $$\rho(x + y) \leq \rho(x) + \rho(y)$$ if $$C$$ is convex.

(c) $$\rho$$ obeys $$\rho(\lambda x) = |\lambda| \rho(x)$$ if $$C$$ is circled/balanced.

(d) $$\{x | \rho(x) < 1 \} \subset C \subset \{x | \rho(x) \leq 1\}$$.

Their proof of Theorem V.1 is:

Let $$\mathscr{U}$$ be a neighborhood base at $$0$$ containing only convex, balanced, absorbing sets; for each $$U \in \mathscr{U}$$, let $$\rho_U$$ be the gauge of $$U$$. By (b) and (c) of the lemma, $$\rho$$ is a seminorm and by (d) the neighborhoods of $$0$$ in the original topology are the same as those in the locally convex topology given by the seminorms $$\{\rho_U | U \in \mathscr{U}\}$$. Since addition is separately continuous in both topologies, the neighborhood about any point are identical in the two topologies.

I have three questions about their proof:

1. How do we know that such a neighborhood base about $$0$$ exists?
2. How does it follow from (d) that the neighborhoods of $$0$$ in the original topology are the same as those in the locally convex topology given by the seminorms $$\{\rho_U | U \in \mathscr{U}\}$$?
3. I do not see how the proof sufficiently proves the forward direction, that if $$V$$ is locally convex then $$0$$ has a neighborhood base of balanced, convex, and absorbing sets. It seems to me that we have only proved this for locally convex spaces induced by a particular family of seminorms, but not in full generality.
• @AnneBauval Thank you for your answer. I am having some trouble seeing how if $U$ contains $\{x\mid\rho_U(x) < 1 \}$ or how the converse ($\{x\mid\rho_U(x) < 1 \}$ containing $\{x\mid\rho_U(x)\le1/2\}\supset\frac12U$) implies that the two topologies are the same. May 29 at 8:14
• Also for (3), how do the sets $\{x\mid\rho(x) < 1 \}$ form a neighborhood base for 0? And how do we know that these sets are necessarily balanced, convex, and absorbing? I apologize for all the questions, I am new to seminorms and locally convex spaces. May 29 at 8:16
• @AnneBauval Reed & Simon define a locally convex space to be a vector space with a family of seminorms separating points. I am still a little confused, but I will think about this proof some more and return to this question later today. Thank you for all your help so far. May 29 at 8:29

2. Because $$U$$ contains $$\{x\mid\rho_U(x) < 1 \}$$ and conversely, $$\{x\mid\rho_U(x) < 1 \}$$ contains $$\{x\mid\rho_U(x)\le1/2\}\supset\frac12U.$$ If two topologies of vector spaces have respectively B and B' as a base of neighborhoods of $$0,$$ they are the same iff every element of B contains some element of B' and conversely, because this is equivalent to having the same neighborhoods of $$0,$$ and in a topological vector space the neighborhoods of $$0$$ determine (by translation) the neighborhoods of any point, hence the topology.
3. Let $$V$$ be a locally convex space, and $$(\rho_i)_{i\in I}$$ be a family of seminorms inducing its topology. To simplify the notations, assume wlog that this family is saturated, i.e. $$\forall i,j\in I\quad\exists k\in I\quad\max\circ(\rho_i,\rho_j)=\rho_k.$$ Then, by definition of the topology on $$V,$$ the "balls" $$\{x\mid\rho_i(x) < c\}$$ for all $$i\in I$$ and all $$c>0,$$ form a base of neighborhoods of $$0.$$ And the "balls" of a seminorm are always balanced, convex and absorbing.