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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ with identity.\nFor a finite-dimensional [[subspace|000AYPA9AJ]] $V \\subseteq A$ and $n \\ge 1$, let $V^n$ be the span of all products of $n$ elements of $V$.\nThe *Gelfand–Kirillov dimension* of $A$ is\n$$\\operatorname{GKdim} A = \\sup_{V} \\, \\limsup_{n \\to \\infty} \\frac{\\log \\dim_k \\big( k + V + V^2 + \\cdots + V^n \\big)}{\\log n},$$\nwhere the supremum runs over all finite-dimensional subspaces $V$ of $A$ and $\\limsup$ is the [[limit superior|0T3RZQXDYD]].\nIt is a number in $[0, \\infty]$.\n\n## Remarks\n\nIf $A$ is finitely generated and $V$ is a finite-dimensional generating subspace with $1 \\in V$, then the supremum is attained at $V$: $\\operatorname{GKdim} A = \\limsup_n \\log d_V(n) / \\log n$, where $d_V$ is the [[growth function|EHY4PYSMGZ]] of $A$.\nSo $\\operatorname{GKdim} A$ depends only on the growth of $A$.\n\nExample: let $A = k[x_1, \\ldots, x_m]$ be a [[polynomial ring|0SDC934R52]] and $V = k + k x_1 + \\cdots + k x_m$.\nThe space $V^n$ is spanned by the monomials of degree at most $n$, so $\\dim V^n = \\binom{n + m}{m}$, which is a polynomial of degree $m$ in $n$.\nHence $\\operatorname{GKdim} A = m$.\n\nExample: if $A$ is finite-dimensional, then $\\dim V^n \\le \\dim A$ for all $n$, so $\\operatorname{GKdim} A = 0$.\n\nNon-example of finiteness: for the free algebra $A = k\\langle x, y \\rangle$ and $V = k + kx + ky$, one has $\\dim V^n = 2^{n+1} - 1$.\nThen $\\log \\dim V^n / \\log n \\to \\infty$, so $\\operatorname{GKdim} A = \\infty$.\n\nThe base of the logarithm does not matter, because only the ratio of two logarithms appears.\n",
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