Index A-measurable, 53 A-partition, 82 absolutely continuous, 107 absolutely continuous function, 152 adjoining the identity, 289 adjoint, 184, 278 aﬃne hyperplane, 246 aﬃne manifold, 246 aﬃne map, 269 aﬃne subspace, 246 Alaoglu, Leonidas, 256 Alaoglu’s Theorem, 256 algebra, 16, 287 algebraically complemented, 232 almost everywhere, 64 almost uniformly, 78 analytic, 293 annihilator, 254 approximate identity, 158 approximate point spectrum, 302 Arzel` a–Ascoli Theorem, 283 Arzel` a, Cesare, 283 Ascoli, Giulio, 283 atom, 87 average, 144 backward shift, 185, 304 Baire, Ren´ e-Louis , 355 Baire Category Theorem, 355 balanced, 240 Banach, Stefan, 23 Banach algebra, 287 Banach limit, 228 Banach space, 23 Banach–Steinhaus Theorem, 235 Banach–Stone Theorem, 280 basis, 175 Bessel, Wilhelm, 177 Bessel’s Inequality, 177 bilateral ideal, 290 bilateral shift, 180 bipolar, 254 Bipolar Theorem, 255 Borel, Emile, 52 Borel function, 53 Borel sets, 52 bounded, 171, 183 bounded linear functional, 37 bounded variation, 1 Bunyakovsky, Viktor, 94 C∗-identity, 319 Cantor, Georg, 13 Cantor function, 61, 62 Cantor middle-third set, 59 Cantor ternary set, 59 Cantor–Lebesgue function, 61 Cauchy, Augustin-Louis, 94 Cauchy’s Theorem, 295 Cauchy–Bunyakovsky–Schwarz Inequality, 94 CBS Inequality, 94 CGT, 231 Chain Rule, 113 Change of Variables Formula, 142 characteristic function, 19 363

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