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  "body": "\n## Summary\n\nA Schauder basis of a [[Banach space|06GZ956657]] is a sequence $(e_n)$ such that every vector is the sum of a unique norm-convergent series $\\sum \\alpha_n e_n$.\nIt replaces the finite linear combinations of a [[basis|0KCQ9Y3QKA]] by infinite series, and so it fits infinite-dimensional spaces.\nThe unit vectors form a Schauder basis of $\\ell^p$ for $1 \\le p \u003c \\infty$ and of $c_0$.\nA Banach space with a Schauder basis is separable, but Enflo showed that not every separable Banach space has a Schauder basis.\n\n## Explanations\n\n### L1 human\n\nIn $\\R^n$ every vector is a combination of the $n$ unit vectors.\nIn a space of infinite sequences one would like to write every vector as an infinite combination $\\alpha_1 e_1 + \\alpha_2 e_2 + \\cdots$ of basic vectors.\nA Schauder basis is a sequence of vectors for which this works, with the coefficients determined uniquely and the series converging in the norm of the space.\nFor example, a sequence $x$ in $\\ell^2$ is the sum of the series $x_1 e_1 + x_2 e_2 + \\cdots$, where $e_k$ has $1$ in place $k$ and $0$ elsewhere.\n",
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