Basically a generalization of both:

- matrix multiplication, with $X=Y=R_{n}$
- dot product, which is a key example of $X =Y$

The key advantage over those concepts however is that it makes more sense in infinite dimensional spaces like Hilbert spaces.

Calling such functions "matrices" in infinite dimensions is stretching it a bit, since we would need to specify infinitely many rows and columns.

The key building block of infinite dimensional linear map is the derivative. In that case, the vectors being operated upon are functions, which cannot therefore be specified by a finite number of parameters.

For example, the left side of the time-independent Schrodinger equation is a linear map. And the time-independent Schrodinger equation can be seen as a eigenvalue problem.

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