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  "body": "\n## Summary\n\nThe Leray–Schauder degree is an integer that counts, with signs, the solutions of $u - K(u) = y$ in a bounded open set of a Banach space.\nHere $K$ is a compact map.\nIt extends the Brouwer degree of maps between finite-dimensional spaces to this infinite-dimensional class.\nIt does not change under compact deformations that keep solutions away from the boundary, and a nonzero degree guarantees a solution.\nThis makes it a main tool to prove existence for nonlinear elliptic equations by continuation from a simpler problem.\n\n## Explanations\n\n### L1 human\n\nIn finite dimensions, the degree of a map counts how many times it covers a point, with orientation signs.\nThe degree survives deformations, so one can compute it on an easy map and transfer the answer to a hard one.\nIn infinite dimensions this fails for general continuous maps, but it works for maps of the form identity minus a compact map.\nSuch maps are close to finite-dimensional ones, because a compact map can be approximated by maps with values in finite-dimensional spaces.\nFor example, $u \\mapsto u - K(u)$ on a ball, with $K$ small and compact, has degree $1$ at $0$, so the equation $u = K(u)$ has a solution in the ball.\n",
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