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  "body": "\n## Summary\n\nA bounded operator between normed spaces is invertible when it has a bounded inverse.\nInvertibility is the basic notion behind the spectrum: a number $\\lambda$ lies in the spectrum of $T$ exactly when $T - \\lambda I$ is not invertible.\nIn infinite dimensions, an injective operator need not be surjective, and a bijective operator needs a theorem to have a bounded inverse.\n\n## Explanations\n\n### L1 human\n\nAn operator is invertible when it can be undone by another bounded operator.\nThink of a change of coordinates that can be reversed without blowing up small errors.\nFor example, the map $x \\mapsto 2x$ on any normed space is invertible, with inverse $x \\mapsto \\tfrac12 x$.\nThe right shift on sequences is not invertible: it is injective but misses every sequence with a nonzero first entry.\n",
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