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@@ -138,7 +138,7 @@
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\item For each $i \in I$, $d_i: E \times E \to [0, \infty)$ defined by $(x, y) \mapsto [x - y]_i$ is a pseudo-metric.
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\item The topology induced by $\seqi{d}$ makes $E$ a topological vector space.
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\item For each $i \in I$, $[\cdot]_i: E \to [0, \infty)$ is continuous.
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\end\{enumerate\}
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\end{enumerate}
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The topology induced by $\seqi{d}$ is the \textbf{vector space topology induced by} $\seqi{[\cdot]}$. In addition,
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\begin{enumerate}
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@@ -160,7 +160,7 @@
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\item If $A$ is convex, then for any $x, y \in E$, $[x + y]_A \le [x]_A + [y]_A$.
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\item If $A$ is circled, then for any $x \in E$ and $\lambda \in K$, $[\lambda x]_A = \abs{\lambda}[x]_A$.
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\item If $A$ is circled, then $\bracs{\rho < 1} \subseteq A \subseteq \bracs{\rho \le 1} \subseteq \ol A$.
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\end\{enumerate\}
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\end{enumerate}
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In particular,
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\begin{enumerate}[start=4]
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@@ -191,7 +191,7 @@
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\item There exists a fundamental system of neighborhoods at $0$ consisting of convex sets.
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\item There exists a fundamental system of neighbourhoods at $0$ consisting of convex, circled, and radial sets.
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\item There exists a family of seminorms $\seqi{[\cdot]}$ that induces the topology on $E$.
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\end\{enumerate\}
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\end{enumerate}
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If the above holds, then $E$ is a \textbf{locally convex} space.
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\end{definition}
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