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In programming, functions transform data. An integer goes in, another comes out. A list is processed into a new one. But what if we had to handle infinite-dimensional data?

Operators in Hilbert space act like functions that take other functions as input and return transformed functions. For example, the differentiation operator takes a function and outputs its derivative. Just as matrices transform vectors, operators manipulate infinite-dimensional data.

Applications range from quantum computing to machine learning. The Hamiltonian in Schrödinger’s equation, kernel methods in data science—both leverage the power of operators. Functions of functions—that’s what Hilbert space operators are.