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