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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a topological vector space over $\\mathbb{K}$.\n$E$ is *locally convex* if every [[neighbourhood|0ZCY0Z80B6]] of $0$ in $E$ contains a [[convex|02AS45T3H9]] neighbourhood of $0$.\n\n## Remarks\n\nEquivalently, the topology of $E$ is the [[topology generated by a family of seminorms|ZVFXQ7S6FY]], namely the family of all continuous seminorms on $E$.\nMany authors also require $E$ to be Hausdorff.\nExample: every normed space is locally convex, since the open balls $\\set{x : \\norm{x} \u003c r}$ are convex and form a neighbourhood base at $0$.\nConvexity of a ball: if $\\norm{x}, \\norm{y} \u003c r$ and $0 \\le t \\le 1$, then $\\norm{t x + (1 - t) y} \\le t \\norm{x} + (1 - t) \\norm{y} \u003c r$.\nNon-example: for $0 \u003c p \u003c 1$, the space $L^p([0, 1])$ with the metric $d(f, g) = \\int_0^1 \\abs{f - g}^p$ is a topological vector space.\nBut its only convex open set containing $0$ is the whole space, so it is not locally convex.\n",
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