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  "body": "\n## Summary\n\nLet $a$ be a normal element of a C*-algebra, and let $f$ be a continuous function on the [[spectrum|EH0CZ19RKP]] of $a$.\nThe continuous functional calculus gives a meaning to $f(a)$.\nIt extends the evaluation of polynomials at $a$ to all continuous functions, and it respects sums, products, adjoints and norms.\nIt is built from the [[commutative Gelfand–Naimark theorem|NT564Y315M]].\nIt is used to define square roots, absolute values, positive and negative parts, and exponentials of operators.\n\n## Explanations\n\n### L1 human\n\nFor a diagonal matrix, applying a function means applying it to each diagonal entry.\nA matrix that can be diagonalized by a unitary change of basis can be handled the same way: diagonalize, apply the function to the eigenvalues, and change back.\nThe continuous functional calculus does this for normal elements of any C\\*-algebra, such as normal operators on an infinite-dimensional Hilbert space.\nFor example, a positive operator has a unique positive square root, given by $f(t) = \\sqrt{t}$.\n",
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