---- [[field|Let]] $\mathbb{F}$ be $\mathbb{R}$ or $\mathbb{C}$. > [!definition] Definition. ([[locally convex space]]) > A [[topological vector space|topological]] [[vector space|vector]] [[topological space|space]] $X$ is said to be **locally convex** if it admits a [[first-countable space|neighborhood basis]] about $0$ consisting of [[balanced set|balanced]] [[convex set|convex sets]]. ^definition > [!basicexample] > - [[norm|Normed spaces]] and [[seminorm|seminormed spaces]] are obviously locally convex, per the equivalence below. ^basic-example > [!equivalence] Seminorm Equivalence. > A [[topological vector space|TVS]] $X$ is locally convex if and only if its [[topological space|topology]] is the [[initial topology|initial topology]] on $X$ with respect to some family of [[seminorm|seminorms]] $\mathcal{P}=\{ p_i: X \to [0, \infty)\}_{i \in I}$. Being an [[initial topology]] (wrt $\mathcal{P}$), we know that $X$ has as [[subbasis for a topology|subbasis]] the collection $\mathscr{S}= \{ p_{i} ^{-1} ([0, \varepsilon_{i}) : \varepsilon_{i}>0, i \in I \}$ and therefore is [[topology generated by a basis|generated by]] the [[basis for a topology|basis]] $\mathscr{B}$ consisting of all finite intersection of sets in $\mathscr{S}$, that is, generated by translates of basic open sets of the form $B_{(\varepsilon_{i_{k}})_{k=1}^{n}}(0)=\{ x \in X : p_{i_{k}}(x)< \varepsilon_{i_{k}} \text{ for all }k=1,\dots, n \} \in \mathscr{B}.$ ^equivalence - [ ] todo fill in the details for proving this equivalence (not hard, i basically did it already in [[weak topology]] ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```