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Spectral Theory Of Operators In Hilbert Space (Paperback)
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Spectral Theory Of Operators In Hilbert Space (Paperback)
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The Present Lectures Intend To Provide An Introduction To The
Spectral Analysis Of Self-Joint Operators Within The Framework Of
Hilbert Space Theory. The Guiding Notion In This Approach Is That
Of Spectral Representation. At The Same Time The Notion Of Function
Of An Operator Is Emphasized. The Definition Of Hilbert Space: In
Mathematics, A Hilbert Space Is A Real Or Complex Vector Space With
A Positive-Definite Hermitian Form, That Is Complete Under Its
Norm. Thus It Is An Inner Product Space, Which Means That It Has
Notions Of Distance And Of Angle (Especially The Notion Of
Orthogonality Or Perpendicularity). The Completeness Requirement
Ensures That For Infinite Dimensional Hilbert Spaces The Limits
Exist When Expected, Which Facilitates Various Definitions From
Calculus. A Typical Example Of A Hilbert Space Is The Space Of
Square Summable Sequences. Hilbert Spaces Allow Simple Geometric
Concepts, Like Projection And Change Of Basis To Be Applied To
Infinite Dimensional Spaces, Such As Function Spaces. They Provide
A Context With Which To Formalize And Generalize The Concepts Of
The Fourier Series In Terms Of Arbitrary Orthogonal Polynomials And
Of The Fourier Transform, Which Are Central Concepts From
Functional Analysis. Hilbert Spaces Are Of Crucial Importance In
The Mathematical Formulation Of Quantum Mechanics.
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