In this chapter, we will learn about:Convergence of vectorsNormExpressions for the normBounds of a self-adjoint transformationMinimax principleConvergence of linear analysisErgodic theoremPower series
![Logo](data:image/svg+xml,<?xml version="1.0" encoding="utf-8" standalone="no"?>%0A<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN"%0A "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd">%0A<svg xmlns:xlink="http://www.w3.org/1999/xlink" width="907.2pt" height="386.374843pt" viewBox="0 0 907.2 386.374843" xmlns="http://www.w3.org/2000/svg" version="1.1">%0A <g id="figure_1">%0A <g id="patch_1">%0A <path d="M 0 386.374843 %0AL 907.2 386.374843 %0AL 907.2 -0 %0AL 0 -0 %0AL 0 386.374843 %0Az%0A" style="fill: none"/>%0A </g>%0A <g id="axes_1">%0A <g id="patch_2">%0A <path d="M 367.323255 30.020542 %0AL 391.543219 30.020542 %0AL 505.645931 212.584882 %0AL 481.425967 212.584882 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="patch_3">%0A <path d="M 367.323255 30.020542 %0AL 391.543219 30.020542 %0AL 277.440506 212.584882 %0AL 253.220542 212.584882 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="patch_4">%0A <path d="M 369.163993 30.020542 %0AL 389.702481 30.020542 %0AL 389.702481 212.584882 %0AL 369.163993 212.584882 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="patch_5">%0A <path d="M 322.381881 111.033468 %0AL 436.484593 111.033468 %0AL 436.484593 131.571957 %0AL 322.381881 131.571957 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="patch_6">%0A <path d="M 367.323255 30.020542 %0AL 391.543219 30.020542 %0AL 448.594575 121.302712 %0AL 424.374611 121.302712 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="patch_7">%0A <path d="M 367.323255 30.020542 %0AL 391.543219 30.020542 %0AL 310.271899 121.302712 %0AL 334.491863 121.302712 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="patch_8">%0A <path d="M 424.374611 121.302712 %0AL 448.594575 121.302712 %0AL 391.543219 212.584882 %0AL 367.323255 212.584882 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="patch_9">%0A <path d="M 334.491863 121.302712 %0AL 310.271899 121.302712 %0AL 367.323255 212.584882 %0AL 391.543219 212.584882 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="patch_10">%0A <path d="M 512.492094 212.584882 %0AL 533.030582 212.584882 %0AL 533.030582 30.020542 %0AL 512.492094 30.020542 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="patch_11">%0A <path d="M 539.876745 212.584882 %0AL 539.876745 30.020542 %0AL 636.864051 30.020542 %0AQ 653.979458 30.020542 653.979458 47.135949 %0AL 653.979458 195.469476 %0AQ 653.979458 212.584882 636.864051 212.584882 %0Az%0AM 560.415233 111.033468 %0AL 624.883266 111.033468 %0AQ 633.440969 111.033468 633.440969 102.475765 %0AL 633.440969 59.116734 %0AQ 633.440969 50.559031 624.883266 50.559031 %0AL 560.415233 50.559031 %0Az%0AM 560.415233 192.046394 %0AL 624.883266 192.046394 %0AQ 633.440969 192.046394 633.440969 183.488691 %0AL 633.440969 140.12966 %0AQ 633.440969 131.571957 624.883266 131.571957 %0AL 560.415233 131.571957 %0Az%0A" clip-path="url(%23p7d9eacca36)" style="fill: %2300a1ff"/>%0A </g>%0A <g id="text_1">%0A <g style="fill: %2300a1ff" transform="translate(27.149522 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-41" d="M 755 0 %0AL 1223 1318 %0AL 3105 1318 %0AL 3573 0 %0AL 4164 0 %0AL 2455 4655 %0AL 1873 4655 %0AL 164 0 %0AL 755 0 %0Az%0AM 1400 1818 %0AL 2146 3918 %0AL 2182 3918 %0AL 2927 1818 %0AL 1400 1818 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-41"/>%0A </g>%0A </g>%0A <g id="text_2">%0A <g style="fill: %2300a1ff" transform="translate(87.183803 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-44" d="M 2000 0 %0AC 3418 0 4218 882 4218 2336 %0AC 4218 3782 3418 4655 2064 4655 %0AL 564 4655 %0AL 564 0 %0AL 2000 0 %0Az%0AM 1127 500 %0AL 1127 4155 %0AL 2027 4155 %0AC 3118 4155 3673 3464 3673 2336 %0AC 3673 1200 3118 500 1964 500 %0AL 1127 500 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-44"/>%0A </g>%0A </g>%0A <g id="text_3">%0A <g style="fill: %2300a1ff" transform="translate(150.459958 312.323656) scale(0.76 -0.76)">%0A <use xlink:href="%23Inter-Regular-41"/>%0A </g>%0A </g>%0A <g id="text_4">%0A <g style="fill: %2300a1ff" transform="translate(213.676738 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-50" d="M 564 0 %0AL 1127 0 %0AL 1127 1700 %0AL 2146 1700 %0AC 3230 1700 3718 2359 3718 3182 %0AC 3718 4005 3230 4655 2136 4655 %0AL 564 4655 %0AL 564 0 %0Az%0AM 1127 2200 %0AL 1127 4155 %0AL 2118 4155 %0AC 2875 4155 3164 3741 3164 3182 %0AC 3164 2623 2875 2200 2127 2200 %0AL 1127 2200 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-50"/>%0A </g>%0A </g>%0A <g id="text_5">%0A <g style="fill: %2300a1ff" transform="translate(275.064768 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-54" d="M 309 4155 %0AL 1773 4155 %0AL 1773 0 %0AL 2336 0 %0AL 2336 4155 %0AL 3800 4155 %0AL 3800 4655 %0AL 309 4655 %0AL 309 4155 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-54"/>%0A </g>%0A </g>%0A <g id="text_6">%0A <g style="fill: %2300a1ff" transform="translate(351.076861 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-49" d="M 1127 4655 %0AL 564 4655 %0AL 564 0 %0AL 1127 0 %0AL 1127 4655 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-49"/>%0A </g>%0A </g>%0A <g id="text_7">%0A <g style="fill: %2300a1ff" transform="translate(397.080829 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-56" d="M 755 4655 %0AL 164 4655 %0AL 1873 0 %0AL 2455 0 %0AL 4164 4655 %0AL 3573 4655 %0AL 2191 736 %0AL 2136 736 %0AL 755 4655 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-56"/>%0A </g>%0A </g>%0A <g id="text_8">%0A <g style="fill: %2300a1ff" transform="translate(461.704796 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-45" d="M 564 0 %0AL 3409 0 %0AL 3409 500 %0AL 1127 500 %0AL 1127 2082 %0AL 3227 2082 %0AL 3227 2582 %0AL 1127 2582 %0AL 1127 4155 %0AL 3373 4155 %0AL 3373 4655 %0AL 564 4655 %0AL 564 0 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-45"/>%0A </g>%0A </g>%0A <g id="text_9">%0A <g style="fill: %23ffffff" transform="translate(583.014294 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-42" d="M 564 0 %0AL 2246 0 %0AC 3346 0 3800 536 3800 1236 %0AC 3800 1973 3291 2373 2864 2400 %0AL 2864 2445 %0AC 3264 2555 3618 2818 3618 3418 %0AC 3618 4100 3164 4655 2191 4655 %0AL 564 4655 %0AL 564 0 %0Az%0AM 1127 500 %0AL 1127 2118 %0AL 2273 2118 %0AC 2882 2118 3264 1709 3264 1236 %0AC 3264 827 2982 500 2246 500 %0AL 1127 500 %0Az%0AM 1127 2609 %0AL 1127 4155 %0AL 2191 4155 %0AC 2809 4155 3082 3827 3082 3418 %0AC 3082 2927 2682 2609 2173 2609 %0AL 1127 2609 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-42"/>%0A </g>%0A </g>%0A <g id="text_10">%0A <g style="fill: %23ffffff" transform="translate(640.459824 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-4f" d="M 4491 2327 %0AC 4491 3800 3627 4718 2436 4718 %0AC 1246 4718 382 3800 382 2327 %0AC 382 855 1246 -64 2436 -64 %0AC 3627 -64 4491 855 4491 2327 %0Az%0AM 3946 2327 %0AC 3946 1118 3282 464 2436 464 %0AC 1591 464 927 1118 927 2327 %0AC 927 3536 1591 4191 2436 4191 %0AC 3282 4191 3946 3536 3946 2327 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-4f"/>%0A </g>%0A </g>%0A <g id="text_11">%0A <g style="fill: %23ffffff" transform="translate(702.115042 312.323656) scale(0.76 -0.76)">%0A <use xlink:href="%23Inter-Regular-4f"/>%0A </g>%0A </g>%0A <g id="text_12">%0A <g style="fill: %23ffffff" transform="translate(767.92651 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-4b" d="M 564 0 %0AL 1127 0 %0AL 1127 1509 %0AL 1709 2164 %0AL 3327 0 %0AL 4009 0 %0AL 2055 2555 %0AL 4009 4655 %0AL 3273 4655 %0AL 1182 2345 %0AL 1127 2345 %0AL 1127 4655 %0AL 564 4655 %0AL 564 0 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-4b"/>%0A </g>%0A </g>%0A <g id="text_13">%0A <g style="fill: %23ffffff" transform="translate(830.12204 312.323656) scale(0.76 -0.76)">%0A <defs>%0A <path id="Inter-Regular-53" d="M 3109 3491 %0AL 3655 3491 %0AC 3630 4193 2982 4718 2082 4718 %0AC 1191 4718 491 4200 491 3418 %0AC 491 2791 946 2418 1673 2209 %0AL 2246 2045 %0AC 2736 1909 3173 1736 3173 1273 %0AC 3173 764 2682 427 2036 427 %0AC 1482 427 991 673 946 1200 %0AL 364 1200 %0AC 418 436 1036 -82 2036 -82 %0AC 3109 -82 3718 509 3718 1264 %0AC 3718 2136 2891 2418 2409 2545 %0AL 1936 2673 %0AC 1591 2764 1036 2945 1036 3445 %0AC 1036 3891 1446 4218 2064 4218 %0AC 2627 4218 3055 3950 3109 3491 %0Az%0A" transform="scale(0.015625)"/>%0A </defs>%0A <use xlink:href="%23Inter-Regular-53"/>%0A </g>%0A </g>%0A </g>%0A </g>%0A <defs>%0A <clipPath id="p7d9eacca36">%0A <rect x="7.2" y="7.2" width="892.8" height="371.974843"/>%0A </clipPath>%0A </defs>%0A</svg>%0A)