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Functional Analysis and its Applications by Amol Sasane

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Functional Analysis and its Applications written by Amol Sasane . This is an other book of mathematics cover the following topics.

  • 1. Normed and Banach spaces
    1.1 Vector spaces
    1.2 Normed spaces
    1.3 Banach spaces
    1.4 Appendix: proof of H¨older’s inequality

  • 2. Continuous maps
    2.1 Linear transformations
    2.2 Continuous maps
    2.2.1 Continuity of functions from R to R
    2.2.2 Continuity of functions between normed spaces
    2.3 The normed space L (X, Y )
    2.4 The Banach algebra L (X). The Neumann series
    2.5 The exponential of an operator
    2.6 Left and right inverses

  • 3. Differentiation
    3.1 The derivative
    3.2 Optimization: necessity of vanishing derivative
    3.3 Optimization: sufficiency in the convex case
    3.4 An example of optimization in a function space

  • 4. Geometry of inner product spaces
    4.1 Inner product spaces
    4.2 Orthogonal sets
    4.3 Best approximation in a subspace
    4.4 Fourier series
    4.5 Riesz representation theorem
    4.6 Adjoints of bounded operators

  • 5. Compact operators
    5.1 Compact operators
    5.2 Approximation of compact operators
    5.3 Appendix A: Bolzano-Weierstrass theorem
    5.4 Appendix B: uniform boundedness principle

  • Bibliography

  • Index

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