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  "body": "\n## Statement\n\nLet $E$ be a real [[vector space|000AYPA9AJ]], let $F \\subseteq E$ be a [[linear subspace|00YC1D652K]], and let $K \\subseteq E$ be a [[convex cone|6VE4KA12VD]].\nLet $\\varphi \\colon F \\to \\R$ be a [[linear form|07YQ1C78G0]] such that $\\varphi(x) \\ge 0$ for every $x \\in F \\cap K$.\nAssume that for every $y \\in E$ there is $x \\in F$ with $y - x \\in K$.\nThen there is a linear form $\\psi \\colon E \\to \\R$ such that $\\psi(x) = \\varphi(x)$ for all $x \\in F$ and $\\psi(y) \\ge 0$ for all $y \\in K$.\n\n## Remarks\n\nThe proof extends $\\varphi$ one dimension at a time, as in the proof of the [[Hahn–Banach theorem|QB5AE3VYZZ]].\nTo add a vector $y \\notin F$, one needs a value $c = \\psi(y)$ with $\\sup \\varphi(x) \\le c \\le \\inf \\varphi(x')$, where $x$ runs over $F$ with $y - x \\in K$ and $x'$ runs over $F$ with $x' - y \\in K$.\nThe hypothesis on $F$ makes both sets nonempty, and positivity of $\\varphi$ on $F \\cap K$ makes the supremum at most the infimum.\nZorn's lemma then gives an extension to all of $E$.\nIn the moment problem, $E$ is a space of functions containing the polynomials and $K$ is the cone of nonnegative functions in $E$.\nThere $F$ is the space of polynomials and $\\varphi$ is the moment functional $x^n \\mapsto m_n$.\nThe positive extension $\\psi$ is then turned into a measure by a representation theorem, which solves the [[Hamburger moment problem|TBNCFXTMMW]] for a positive moment functional.\nWithout the assumption that $y - x \\in K$ for some $x \\in F$, an extension may fail to exist.\nFor example, take $E = \\R^2$, $K = \\set{(s, t) : t \u003e 0} \\cup \\set{(s, 0) : s \\ge 0}$, $F = \\R \\times \\set{0}$ and $\\varphi(s, 0) = s$; then $\\varphi \\ge 0$ on $F \\cap K$, but a positive extension $\\psi(s, t) = s + ct$ would need $s + c \\ge 0$ for all $s$, which is impossible.\n\n## Explanations\n\n### L1 human\n\nA linear rule that gives nonnegative values on the \"positive\" vectors of a subspace can be extended to the whole space without losing this property.\nThe only condition is that the subspace is large enough: every vector must lie above some vector of the subspace.\nIt is the order version of the Hahn–Banach theorem.\n\n### L2 human\n\nThe theorem is the abstract tool behind the existence parts of moment problems.\nA sequence $(m_n)$ defines a linear form $L$ on polynomials by $L(x^n) = m_n$.\nSuppose that $L(p) \\ge 0$ for every polynomial $p \\ge 0$.\nThen the theorem extends $L$ to a positive linear form on a larger space of functions, such as the continuous functions bounded by a polynomial.\nThe [[Riesz representation theorem|TTC40N49M6]] then gives a measure with moments $m_n$.\nThe same argument, with polynomials in several variables and nonnegativity on a closed set, gives the multivariate form of this existence result.\n",
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