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Functional Analysis

Banach and Hilbert spaces, operators, spectral theory.

Hvad vi kan lære

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.

Flere lektioner

Spaces of functions

Functional analysis Normed vector space Banach space Hilbert space Inner product space Complete metric space Sequence space Orthonormal basis Separable space

Operators

Bounded operator Linear map Dual space Compact operator Adjoint Self-adjoint operator Unitary operator Operator norm

The big theorems

Hahn–Banach theorem Open mapping theorem (functional analysis)Closed graph theorem Uniform boundedness principle Spectral theorem Riesz representation theorem Weak topology Banach fixed-point theorem

Fourier and applications

Fourier series Fourier transform Convolution Sobolev space Distribution (mathematics)

Nogle af disse erfaringer er tilpasset fra åbne tekster: Wikipedia. Hver lektionsside angiver kilden og licensen.

Symboler brugt her

Tryk på ethvert symbol for den fulde definition, et billede, og hvad hvert bogstav i det betyder.

Spørgsmål folk spørger

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.

Har du et problem?
Skriv det eller træk det på løsersiden: hvert skridt, et billede, svaret skjult, indtil du spørger. Løs →

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