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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$, and let $V$ be a [[vector space|05EGHEQZAW]] over $\\mathbb{K}$.\nA subset $A \\subseteq V$ is *absorbing* if the following holds.\nFor every $x \\in V$ there is $r \u003e 0$ such that $x \\in c A$ for every $c \\in \\mathbb{K}$ with [[absolute value|00S7C68D51]] $\\abs{c} \\ge r$.\n\n## Remarks\n\nAn absorbing set $A$ contains $0$: for $x = 0$ there is $c \\ne 0$ with $0 \\in cA$, so $0 = c a$ for some $a \\in A$, and then $a = 0$.\nIf $A$ is [[balanced|2TDSHJJYS4]], then $A$ is absorbing if and only if for every $x \\in V$ there is $t \u003e 0$ with $t x \\in A$.\nExample: the open unit ball $A = \\set{x : \\norm{x} \u003c 1}$ of a normed space is absorbing; for $x \\in V$ take $r = \\norm{x} + 1$, and then $\\norm{x / c} \u003c 1$ whenever $\\abs{c} \\ge r$.\nNon-example: in $\\R^2$, the axis $A = \\R \\times \\set{0}$ is not absorbing, because $(0, 1) \\notin c A$ for every $c$.\nIn a topological vector space every neighbourhood of $0$ is absorbing, since $c \\mapsto c^{-1} x$ is continuous and tends to $0$.\n",
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