# From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ?

I'm thinking of a text that goes through a basic introductory book (the more known it is, the better) on real analysis - like Principles of Mathematical Analysis by Rudin - and explains which theorems from real analysis still hold in the setting of Hilbert spaces and which will fail. (A discussion for Banach spaces would also be ok - and I'm also interested in this case - but right now the case for Hilbert spaces is more pressing) To be more precise, I would like to know which results from chapters 5 (Differentiation), 6 (The Riemann-Stieltjes Integral), 7 (Sequences and Series of Functions), 9 (Functions of Several Variables) and 11 (The Lebesgue Integral) generalize as just described.
Of course Rudins book was just an example of a book which I used, because the book is very known. Any solid (but not-too-graduate-level) analysis book, which treats functions $\mathbb{R}\rightarrow \mathbb{R}$ and $\mathbb{R}^n\rightarrow \mathbb{R}^m$, has chapters like those which cover pretty much the same basic theorems.
To give an example of what I already know: The generalization of chapter 4 to Hilbert spaces I know already. In functional analysis the concept of continuity is extended from functions $\mathbb{R}\rightarrow \mathbb{R}$ to mappings $\mathcal{H}\rightarrow \mathcal{H}$ (or more general: metric spaces).
Which results can be extended to which setting ($\mathcal{H}\rightarrow \mathcal{H}$ or $\mathbb{R}\rightarrow \mathcal{H}$) depends on the concept of course: Derivatives can be extended to the first case, integrals only to the second (as far as I know)).
To give a further example: A lot of theorem for derivatives between Hilbert/Banach spaces have a proof that is the same as the proof of the theorem in the case $\mathbb{R}\rightarrow \mathbb{R}$. I'm interested exactly in these theorems -- and want to know which theorems (maybe with the helpt of a counterexample, if the theorem would even make sense in the abstract setting) can't be adapted like this to the more abstract setting (what the "right version" of that $\mathbb{R}\rightarrow \mathbb{R}$ theorem - that fails when "mechanically" extended to $\mathcal{H}\rightarrow \mathcal{H}$ - would be in the abstract setting doesn't interest me at this point).

(I know of course that there are books that develop the whole analysis for Hilbert/Banach spaces in the first place - most graduate analysis books seem to do that - but I don't want to go through the whole book and look at every proof to see if it goes through in the case of Hilbert/Banach spaces)

EDIT: Here's more precise description of what I want, in case there is confusion. I shall describe what I'm looking for for brevity only for the concept of the Lebesgue-integral. But the same holds for all of the other basic concepts in analysis (like the derivative of functions $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$, all of the topological properties of $\mathbb{R}^n$ etc.).
Let $L\mathbb{K}$ denote the set of theorems which hold for the L-integral of a function $f:X\rightarrow\mathbb{K}$, where $\mathbb{K}=\mathbb{R},\mathbb{C}$ and $X$ is a measurable space.
Similarly let $L\mathcal{H}$ denote the set of theorems which hold for L-integrals of functions $f:X\rightarrow\mathcal{H}$, where $\mathcal{H}$ is a, say, Hilbert space.
Now I'm not interested in knowing all theorems from $L\mathcal{H}$, since they probably are rather complicated. I'm only interested in those theorems/definitions from $L\mathcal{H}$ whose proof consists only of a "mechanical" replacement of all occurences of "$\mathbb{K}$" with "$\mathcal{H}$", i.e. the theorems from $L\mathbb{K}$ which generalize immediatly to $L\mathcal{H}$.
That there may still be some theorem $t_{\mathcal{H}}$ in $L\mathcal{H}$, that is the "appropriate generalization" of $t_{\mathbb{K}}$ from $L\mathbb{K}$, but whose proof and form divergence significantly from those of $t_{\mathbb{K}}$ does not interest me.
Additionally I would like to know which theorems from $L\mathbb{K}$ fail - or cannot even be formulated - when we replace "$\mathbb{K}$" with "$\mathcal{H}$", i.e. which fail when "straightforward generalized".

• As Hilbert/Banach spaces are vector spaces one should ask about the differences of finite and infinite dimensional vector spaces. – Michael Hoppe Oct 31 '13 at 10:10
• Why do you need the text to be longer than 30 pages? – Rob Arthan Nov 2 '13 at 1:03
• @RobArthan Sorry, I meant not longer. I fixed it now. – temo Nov 2 '13 at 17:03
• I thought a little a while if I could give a more elaborate answer to your edited question. I think your question has to be focused much more. It doesn't help to put a bounty on a question which isn't thought wellthrough. It's like asking: "Tell me about the cosmos, but don't give me references that only partially cover what I ask for". How long are the standard texts in analysis, topology and measure theory you have read so far? – Marc Palm Nov 4 '13 at 7:40
• The following is not within 30 pages, but it comes closest to fitting what I think you want than any book I know: Leopoldo Nachbin, Introduction to Functional Analysis: Banach Spaces and Differential Calculus, Monographs and Textbooks in Pure and Applied Mathematics #60, Marcel Dekker, 1981, xii + 166 pages. – Dave L. Renfro Nov 4 '13 at 20:48

Try combining Rudin's book with another book called Applied Analysis by John Hunter. Chapter 5 gives great insight on Banach Spaces and Chapter 6 is dedicated solely to Hilbert Spaces. Dr. Hunter actually has his book free on this website:

Have you tried this book called Differential Calculus, by H. Cartan?

It develops Calculus from the very beginning always in the context of normed or Banach spaces, starting from some elementary functional analysis.

For instance, in Chapter $1$, Section $1$ is about norms, linear and multilinear mappings, continuity, etc.. all in normed/Banach spaces.

Section $2$ then develops the basics of differentiation in this setting.

Section $3$ is about mean value theorems and applications.

Later chapters develop Inverse/Implicit function theorems, higher order derivatives (as multilinear maps), Taylor's formula, etc.

It is very thorough, though, in that it goes all the way into the theory of ODEs in Banach spaces.

At least for Hilbert spaces, you can recover every thing from the scalar-valued measures simply by using $v = \sum \langle e,v \rangle e$, where $e$ runs through an orthonormal basis.

So it's possible to define $$\int\limits_{X} f(x) d x := \sum \int\limits_{X} <e,f(x)> d x \cdot e$$ if the convergence on the right hand side is absolute. But crucial results like monoton convergence and Lebesgue dominated convergence are probaly not available. What is monoton convergence here?

For Banach spaces, things seem to get more subtle, but a theorem of Pettis describe measurability of a Banach-valued function as being equivalent to measurability of the composition with all functionals.

Here are some further references: https://www.encyclopediaofmath.org/index.php/Bochner_integral

• This seems to be at a rather advanced level. I think I'm looking for something more basic, but more detailed (also including things like derivatives etc.). Thanks nonetheless for these infos. – temo Oct 31 '13 at 18:50
• If you don't know about Lebesgue integrals, you should learn that first. In general, most can be reduced to calculus on $\mathbb{C}$-valued function by applying functionals, say taking derivatives etc, but there are some subtleties. Don't expect an easy general theory. – Marc Palm Nov 1 '13 at 7:59
• There seems to be a misunderstanding here - or I am misunderstanding something ? Anyway, I wrote an "Edit" section to my question to state more clearly what I want, since you seem to think that I want to know all (!) of $L\mathcal{H}$. What I mean by $L\mathcal{H}$ is explained in the "Edit". (I do have a - though rather superficial - acquaintance with L-integrals.) – temo Nov 1 '13 at 17:42