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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $E$ be a vector space over $\\mathbb{K}$.\nSuppose $E = \\bigcup_{n \\ge 1} E_n$, where $E_1 \\subseteq E_2 \\subseteq \\cdots$ are linear subspaces.\nSuppose each $E_n$ carries a topology making it a [[Fréchet space|3GAA614673]], and each inclusion $E_n \\to E_{n+1}$ is continuous.\nThe *inductive limit topology* on $E$ is the locally convex topology defined as follows.\nA set is a [[neighbourhood|0ZCY0Z80B6]] of $0$ if it contains a [[convex|02AS45T3H9]] [[balanced|2TDSHJJYS4]] set $U$ with the following property.\nFor every $n$, the set $U \\cap E_n$ is a neighbourhood of $0$ in $E_n$.\nA [[locally convex space|B6GY2KJMWW]] is an *LF-space* if it is Hausdorff and arises in this way.\n\n## Remarks\n\nThe inductive limit topology is the finest locally convex topology on $E$ for which all inclusions $E_n \\to E$ are continuous.\nMany authors require the limit to be *strict*: each $E_n$ carries the topology induced by $E_{n+1}$.\nConsider a strict limit with each $E_n$ closed in $E_{n+1}$.\nThen $E$ is Hausdorff and complete, and it induces the given topology on each $E_n$.\nA set is bounded in $E$ if and only if it lies in some $E_n$ and is bounded there.\nExample: let $\\Omega \\subseteq \\R^d$ be open, and let $K_1 \\subseteq K_2 \\subseteq \\cdots$ be compact sets with union $\\Omega$, each inside the interior of the next.\nLet $\\mathcal{D}_{K_n}$ be the Fréchet space of smooth functions supported in $K_n$.\nThen $C_c^\\infty(\\Omega) = \\bigcup_n \\mathcal{D}_{K_n}$ is a strict LF-space.\nNon-example: the space $c_{00}$ of finitely supported sequences with the sup norm is not an LF-space, since every LF-space is barrelled and this space is not.\n",
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