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Functional Analysis

Banach and Hilbert spaces, operators, spectral theory.

ບົດຮຽນ

Advanced Normed and Banach spaces Vector spaces with a length; completeness; the classical sequence and function spaces. sqrt(3^2 + 4^2) Advanced Hilbert spaces and the spectral theorem Inner products, orthonormal bases, Fourier series, and eigenvalues of operators. eigenvalues of [[2,1],[1,2]]

Linear algebra in infinitely many dimensions, where functions are the vectors. Quantum mechanics, PDEs and signal processing all live here; the spectral theorem is the eigenvalue story told for operators.

ບົດຮຽນ​ເພີ່ມເຕີມ

Spaces of functions

Functional analysisNormed vector spaceBanach spaceHilbert spaceInner product spaceComplete metric spaceSequence spaceOrthonormal basisSeparable space

Operators

Bounded operatorLinear mapDual spaceCompact operatorAdjointSelf-adjoint operatorUnitary operatorOperator norm

The big theorems

Hahn–Banach theoremOpen mapping theorem (functional analysis)Closed graph theoremUniform boundedness principleSpectral theoremRiesz representation theoremWeak topologyBanach fixed-point theorem

Fourier and applications

Fourier seriesFourier transformConvolutionSobolev spaceDistribution (mathematics)

ບົດຮຽນບາງບົດນີ້ ແມ່ນໄດ້ນຳເອົາຈາກບົດຂຽນເປີດ: Wikipedia. ທຸກໆ​ໜ້າ​ບົດຮຽນ​ມີ​ຊື່​ແຫຼ່ງ​ຂອງ​ມັນ ແລະ ໃບອະນຸຍາດ​ຂອງ​ມັນ.

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ຄໍາຖາມທີ່ຄົນຖາມ

What is a Hilbert space?

A vector space with an inner product (so lengths and angles make sense) that is complete (no missing limit points). Square-integrable functions form one; quantum states live in one.

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