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  "body": "\n## Definition\n\nLet $\\mathbb{K}$ be $\\R$ or $\\C$.\nA *Banach algebra* over $\\mathbb{K}$ is a [[normed algebra|7VS7JZ3Y5B]] $(A, \\norm{\\cdot})$ over $\\mathbb{K}$ such that the normed space $(A, \\norm{\\cdot})$ is complete, that is, a [[Banach space|0BDTTM24DP]].\nSo $A$ is an associative $\\mathbb{K}$-algebra with a complete norm such that $\\norm{xy} \\le \\norm{x}\\norm{y}$ for all $x, y \\in A$.\nA Banach algebra is *unital* if it has a unit $1$, and then one usually also asks $\\norm{1} = 1$.\n\n## Remarks\n\nExample: for a compact Hausdorff space $X$, the algebra $C(X)$ of continuous functions $X \\to \\C$ with $\\norm{f} = \\sup_{x \\in X} \\abs{f(x)}$.\nSubmultiplicativity holds since $\\abs{f(x) g(x)} \\le \\norm{f}\\norm{g}$ for every $x$, and completeness holds because a uniform limit of continuous functions is continuous.\nExample: the bounded linear operators on a Banach space with the operator norm.\nNon-example: the polynomial functions on $[0, 1]$ with the same sup norm form a normed algebra but not a Banach algebra.\nThey are dense in $C([0, 1])$ by the Weierstrass approximation theorem, and $e^x$ is a uniform limit of polynomials that is not a polynomial, so completeness fails.\nIn a unital Banach algebra, $1 - x$ is invertible with inverse $\\sum_{n \\ge 0} x^n$ when $\\norm{x} \u003c 1$.\n",
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