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  "body": "\n## Summary\n\nA finite-rank operator is a bounded linear map between normed spaces whose image is finite-dimensional.\nSuch operators behave like matrices, even when the spaces are infinite-dimensional.\nThey are the simplest operators of functional analysis, and many other classes, such as compact and trace class operators, are built as limits of them.\n\n## Explanations\n\n### L1 human\n\nA finite-rank operator takes a possibly huge space and squeezes it into a finite-dimensional piece.\nPicture a projector that casts every object of a large room onto one flat wall: the shadows live in a small space.\nFor a concrete example, on the space of square-summable sequences, the map $(x_1, x_2, x_3, \\ldots) \\mapsto (x_1 + x_2, 0, 0, \\ldots)$ has image the multiples of the first unit vector.\nIts image has dimension $1$, so it is an operator of rank $1$.\n",
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