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  "body": "\n## Summary\n\nA Fréchet algebra is an associative algebra that is also a [[Fréchet space|3GAA614673]], with a jointly continuous multiplication.\nIt generalizes a Banach algebra: the topology comes from a countable family of [[seminorms|FP5GVCN8GV]] instead of a single norm.\nAlgebras of smooth functions, of holomorphic functions, and of continuous functions on the real line are Fréchet algebras but not Banach algebras.\n\n## Explanations\n\n### L1 human\n\nSometimes no single norm can measure the functions of interest, for example all continuous functions on the real line, which may be unbounded.\nInstead one measures a function by its maximum on $[-1, 1]$, on $[-2, 2]$, on $[-3, 3]$, and so on.\nA sequence converges when it converges uniformly on each of these intervals.\nWith this notion of convergence, the continuous functions on $\\R$ form a Fréchet algebra.\n",
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