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  "body": "\n## Summary\n\nThe Rademacher functions $r_1, r_2, \\dots$ on $[0, 1)$ are square waves that record the binary digits of a point.\nThe value $r_k(t)$ is $1$ or $-1$ according as the $k$-th binary digit of $t$ is $0$ or $1$.\nThey form an orthonormal system in $L^2([0, 1))$ which is not complete.\nFor [[Lebesgue measure|2Z2QVWZ262]] they are independent random signs, so they model coin tossing inside analysis.\nTheir finite products are the [[Walsh functions|F12A5QKW8Z]].\n\n## Explanations\n\n### L1 human\n\nSplit $[0, 1)$ into halves and put $+1$ on the left half and $-1$ on the right: that is $r_1$.\nSplit into quarters and alternate $+1, -1, +1, -1$: that is $r_2$.\nEach next function flips twice as fast.\nIf one picks $t$ at random, the values $r_1(t), r_2(t), \\dots$ behave like fair coin tosses, one for each binary digit of $t$.\n",
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