I am trying to learn more about function spaces, properties and structure of the functions in each space but also properties and structures of the function spaces themselves, with different metrics maybe, embeddings and dense embeddings (with the classical or even different metrics for each space), and approximation arguments for each space. This would include, spaces of functions on manifolds,on $\mathbb{R}$, on $\mathbb{R}$, on $\mathbb{R}^n$, on $\mathbb{C}^n$, on subsets of manifolds or $\mathbb{R}^n$ or $\mathbb{C}^n$, even more general metric or topological sets. Spaces of differential functions, of continuous functions (even almost everywhere differential or alamost everywhere continuous etc.), integrable functions, covex functions, real analytic functions, holomorphic functions, Sobolev spaces etc.
It would be really nice, if your recommendation are from the more classical to the more general case. For example I think it would be more constructive for me if I'd start with some more classical spaces, for example spaces on $\mathbb{R}^n$ and then go to manifolds. But my reasoning for that could be of course false. Furthermore, it could be books, lecture notes or whatever youy may think would be helpful.
Lastly, the more theorems, propositions, properties etc. the better, but I would also like to see many examples, useful methods and tricks, intuitional approaches and explanations and many exercises, if possible. Of course not only the "trivial" ones.
I have basic (to maybe somewhat advanced knowledge for some of them) knowledge of one-dimensional analysis, multivariable analysis, cmplex analysis, measure theory, functional analysis, Fourier analysis, Banach algebras, operator theory, spectral theory, Sobolev spaces etc.
Some books I have already in mind are for example,
- Differential Topology, Hirsch,
- Theory of Function Spaces, Triebel,
- Measure Theory and Fine Properties of Functions, Evans & Gariepy
It is probably much to ask and maybe very vague what I am looking for but any help and recommendations are welcome. Thank you!
Best Answer
So many lovely books on these and related topics. A few favorites:
Matousek, Lectures on discrete geometry (esp. Chap 15 on embeddings)
Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations
Katznelson, Harmonic analysis
Helson, Harmonic analysis
Rudin, Fourier analysis on groups
Lindenstrauss and Tzafriri, Classical Banach spaces
Kahane, some random series of functions
Daubechies, Ten Lectures on Wavelets
Pisier The Volume of Convex Bodies and Banach Space Geometry
See also: Wells and Williams, Embeddings and Extensions in Analysis