Tagged Questions
2
votes
2answers
97 views
Banach Algebra counterexample
Can someone give me an exemple of a Banach Algebra $\mathbb{A}$, for which there is no isometric representation in a Hilbert Space ? (with proof or references to proof)
Thank you very much :)
2
votes
1answer
34 views
Gelfand-Naimark Theorem with separable algebras
If a C$^{\star}$- algebra is separable, is there a representation in a ,also separable, Hilbert space ? Probably it's not hard to adapt the proof of the Gelfand-Naimark Theorem, but can someone give ...
1
vote
1answer
70 views
A sequence of integral tends to zero
Suppose that $a<b$ are two fixed real numbers, and $g_n$ is a sequence of real functions on $[a,b]$.
What about good conditions approximately equivalent to the following proposition:
...
5
votes
3answers
61 views
Introductory/Intuitive Functional Analysis Book
Can you recommend a gentle introduction to the abstract thinking and motivation of functional analysis? I'm looking for a book that holds you by the hand and shows the details of exercises, etc.
...
3
votes
1answer
43 views
Dual of holomorphic functions (with the $L^1$ topology)
Let $\Omega$ be a connected domain of the complex plane, and let $E$ be the vector space of integrable holomorphic functions on $\Omega$. Then it can be checked that $E$ is a closed subspace of ...
5
votes
0answers
72 views
Solution to $\Delta_g u = \delta-1$ on a 2-sphere.
Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
5
votes
1answer
60 views
Zeta Regularized Determinant of Laplacian
Can anyone point me to a resource where the zeta regularized determinant of the Laplacian is explicitly computed for simple two dimensional surfaces, say a rectangle or torus or cylinder?
5
votes
0answers
205 views
Problem of Scottish Book
Does anyone know if the problem 50 to Banach written in The Scottish Book is resolved? The problem is:
Prove that the integral of denjoy is a Baire functional in the space M ( that is to say, in ...
4
votes
1answer
93 views
Discreteness of eigenvalues for certain operators - can this approach be made rigorous?
I was idly thinking about why one might naïvely expect a discrete spectrum of eigenvalues for a linear operator $L$ when I dreamt up the following argument (which I expect isn't new instead - ...
4
votes
1answer
106 views
Quotients of C*-algebras
It is known that every unital separable C*-algebra is a quotient of the full group C*-algebra $C^*(F_I)$, where $F_I$ is the free group generated by some index set $I$.
Can we drop the ...
0
votes
2answers
50 views
a version of taylor theorem
I need to a version Taylor theorem (Taylor expantion) in $H^2(0,1)$.
What is the difference between it and usual Taylor expantion.
Where can I find it?
Thanks.
3
votes
1answer
232 views
Sets $f_n\in A_f$ where $f_{n+1}=f_n \circ S \circ f^{\circ (-1)}_n$ and operator $\alpha(f_n)=f_{n+1}$
Let's start with a function on the Reals (in this case for $x=0$ is not defined): for example $f(x)=b/x$, $x \in \mathbb R$
I define:
$$f_0:=f$$
$$f_{n+1}:=f_n \circ S \circ f^{\circ ...
0
votes
1answer
60 views
Is Conway's “Course in Functional Analysis” suitable for self-studying?
Is John B. Conway's book "A Course in Functional Analysis" a good book for self-studying functional analysis?
(I have a solid knowledge of undergraduate analysis and linear algebra, group theory, ...
10
votes
1answer
210 views
Real analysis textbok that develops the subject in a self-motivated, coherent fashion?
Well, it seems as though I just failed my analysis prelim for the second time... I have one more try in about $5$ months.
I'm failing to build up a framework for how to think about analysis problems. ...
2
votes
0answers
81 views
Books for Practice Problems
I'm not asking for the solution to the question posted below, but instead for references to books where I can find similar questions.
I'd be grateful for any advice you can give. Thank you.
0
votes
0answers
31 views
Functional Analysis- Albert Wilanksy
Has anyone used this book? Can anyone recommend it for self-study? Thanks
1
vote
1answer
54 views
Reference Request: Vector Spaces
I am a new student in the field of functional analysis. I'm looking for references that make sense for all kinds of vector spaces, such as the difference between $L^2$ and $l^2$ and others like: ...
6
votes
1answer
120 views
Short and elegant introduction to Sobolev spaces
I am preparing a course on Nonlinear Analysis, and I need to teach the most important facts about Sobolev spaces to my students. I know most books on this subject, from Brezis' to Adams', from Mazya's ...
0
votes
0answers
99 views
exercise of Functional Analysis in Applied Mathematics and Engineering
I want to solve the exercises in Functional Analysis in Applied Mathematics and Engineering by Pedersen. Some of problems are very hard. Do you know any book that solves the exercise of this book? ...
3
votes
1answer
180 views
Is “Functional Analysis” by “Yosida” a good book for self study?
I was wishing to start studying by myself the book Functional Analysis by Yosida, does anyone have already used it, is it a good reference?
0
votes
1answer
135 views
Functional Analysis - Where to go from here?
The short version of this question is this:
I like functional analysis and want to learn more. I've taken a class on it and I've read the books by Brezis and Conway. Where can I go from here? Do
...
2
votes
0answers
107 views
Existence of solution of PDE using Galerkin method
I wonder if anyone can give me a reference to a paper/book that rigorously addresses how to use the Galerkin method to show existence/uniqueness of a PDE. The usual suspects (Evans, Renardy, ...) do ...
11
votes
2answers
207 views
How are infinite-dimensional manifolds most commonly treated?
At a summer school I recently attended, infinite-dimensional manifolds popped up. I have never worked with them before (although I'm very familiar with finite-dimensional manifolds). The lecturer at ...
7
votes
1answer
126 views
Dual space of space of all smooth function
On the space $C^\infty(S^1,\mathbb R)$, for each $n\in \mathbb N$, define
$$p_N(\gamma)= \max\{|f^{(k)}(t): t\in S^1, k\leq N\}$$
Topology of all norms above define a metrizable locally convex ...
2
votes
1answer
72 views
A reference request for sums of $C^*$-algebras
Does anyone know where I can find a reference for the following well-known fact:
Let $(X_i)_{i\in I}$ be a family of compact Hausdorff spaces and let $X$ be the disjoint sum of all $X_i$s.
Then
...
4
votes
0answers
136 views
Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.
Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$.
The following is from the book "Sobolev spaces" ...
5
votes
1answer
88 views
Invertibility of laplacian operator
Let $\Omega\in\mathbb{R}^n$ be a bounded open set with smooth boundary. How to prove the invertibility of $$- \triangle:H^2_0(\Omega) \to L²(\Omega) $$
The injectivity is easy. But how to prove ...
40
votes
3answers
911 views
Paul Erdos's Two-Line Functional Analysis Proof
Legends hold that once upon a time, some mathematicians were rather pleased about a 30-ish page result in functional analysis. Paul Erdos, upon learning of the problem, spent ten or so minutes ...
9
votes
2answers
279 views
problem books in functional analysis
There are many excellent problem books in real analysis.I'm looking for a problem book in functional analysis or a book which contains a lot of problems in functional analysis (Easy and hard problems) ...
3
votes
1answer
72 views
Does $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for bounded operators on Hilbert space?
If $A$ is a bounded linear operator on a Hilbert space $H$ is it true that $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for all $x\in H$? If not, can we at least establish inequality in one ...
3
votes
1answer
122 views
Functional analysis summary
Anyone knows a good summary containing the most important definitions and theorems about functional analysis.
2
votes
0answers
74 views
A question about a metric on $\mathbb{R}^\mathbb{N}$
Consider the metric space $(\mathbb{R}^{\mathbb{N}},d)$ where for $x,y\in\mathbb{R}^\mathbb{N}$
$$ d(x,y) = \sum_{n=1}^{\infty} 2^{- n} \frac{\bigvee_{k\leq n}\left|x_k-y_k\right|}{1 + \bigvee_{k\leq ...
4
votes
3answers
187 views
(Product) Lebesgue measure on infinite dimensional spaces?
I am trying to understand measure construction procedures on infinite-dimensional spaces. Why is it not possible in general to construct Lebesgue measure on $\mathbb{R}^\mathbb{N}$ or ...
1
vote
0answers
37 views
A reference to basic properties of compact operators without assuming completeness
I am looking for a textbook showing that (i) every compact operator is bounded and (ii) composing a compact operator with a bounded one (and a bounded operator with a compact one) gives a compact ...
1
vote
0answers
49 views
Containment of an element to an operator system
This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
1
vote
0answers
57 views
Positive maps on $\mathcal{B}(\mathcal{H})$ to itself
Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the ...
3
votes
2answers
196 views
Exercise books on functional analysis
I have known a lot of excellent textbooks on functional analysis:
Functional Analysis (Walter Rudin)
Functional Analysis, Sobolev Spaces and Partial Differential Equations (Haim Brezis)
+....
I ...
1
vote
1answer
72 views
Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$
I'm looking for articles describing (or proving nonexistence) of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$.
Since $\ell_q^m$ is finite ...
3
votes
1answer
283 views
Are smooth functions with compact support weakly-* dense in $L^\infty$?
My question is this : given $f \in L^\infty(\mathbb{R}^2)$, can we find a sequence $\phi_n$ of smooth, compactly supported functions (test functions) such that for any $g \in L^1(\mathbb{R}^2)$,
...
0
votes
0answers
55 views
Time dependent or Bochner space references?
Does anyone have any recommendations where I can learn about time dependent or Bochner spaces? I mean spaces like $L^p(0,T; H^{-1}(\Omega))$. I think one needs some knowledge of distributions, so any ...
1
vote
1answer
87 views
Morrey space and Campanato space.
I'd like to know a lot about Morrey space and Campanato spaces. For example, I'd like to know how can I see the details presents here. I'd like some reference about this. I thank you very much.
1
vote
0answers
34 views
geometric charaterization of complex interpolation spaces $(H,Y)_\theta$ where $H$ is a Hilbert space?
Let $C$ be the class of Banach spaces $X$ such that there exists $0<\theta<1$, a Hilbert space $H$ and a Banach space $Y$ such that
$$
X=(H,Y)_\theta
$$
(complex interpolation of Calderon).
...
2
votes
1answer
113 views
Extension of Choi's theorem on extreme completely positive maps
In this paper Man-Duen Choi gave a criteria for a completely positive map to be extreme. For convenience I am writing it below.
Let $\phi:\mathcal{M}_n\rightarrow\mathcal{M}_m$. Then $\phi$ is
...
7
votes
1answer
238 views
$C_0(X)$ is not the dual of a complete normed space
Let $X$ be any locally compact Hausdorff space and assume that it is not compact.
I've heard that the Banach space $(C_0(X),\|\!\cdot\!\|_\infty)$ is not isometrically isomorphic to the (norm) dual of ...
2
votes
0answers
50 views
What is a good resource for functional derivatives and functional determinants?
What is a good resource for functional derivatives, functional determinants, etc.? What is the branch of mathematics dealing with those things? It is not in my functional analysis book. What is a ...
3
votes
0answers
78 views
Necessary and sufficient conditions for $G$ to have $A(G)$ isomorphic to an operator algebra?
Let $G$ be a locally compact group. We denote by $A(G)$ the Fourier algebra of $G$.
An operator algebra is a closed subalgebra of $B(H)$ where $H$ is a Hilbert space.
What are the necessary and ...
0
votes
0answers
27 views
Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega ...
3
votes
1answer
68 views
Reference for: $G$ discrete iff the measure algebra $M(G)$ is weakly amenable.
I search the reference for the proof of the following theorem:
Let $G$ be a locally compact group. Then
the group $G$ is discrete if and only if the measure algebra $M(G)$ is weakly amenable.
The ...
2
votes
0answers
47 views
Good references on Distribution Theory [duplicate]
Possible Duplicate:
Distribution theory book
Two books I have been reading are Strichartz's A Guide to Distribution Theory and Fourier Transforms and PartII of Rudin's Functional Analysis . ...
1
vote
1answer
103 views
Discrete Sobolev Space and Sobolev Spaces of Banach Space valued functions
This is a reference request.
Can someone kindly give me some refernce(Books/papers) on
Discrete Sobolev Space (like we use Discrete $L^p$ spaces of $g\colon\Omega\to\Bbb R $ maps with norm given as ...
