# Functional Analysis

Is there any metric in which the set of real numbers R is compact
I want to check those metric's in which the set $R$ of real numbers is compact!! Note that if that metric is induced from a norm, then R with that metric is not compact
Geoff Diestel · Texas A&M University System
The metric d(x,y)=|arctan(x)-arctan(y)| is a compact metric on the extended real number line where plus and minus infinity are viewed as actual endpoints of the space. In this metric, every unbounded sequence has a convergent subsequence. Thus, every sequence has a convergent subsequence. This is sequential compactness but that is equivalent to compactness in metric spaces. Removing plus and minus infinity makes this metric incomplete. I don't know off the top of my head an explicit compact metric for just the real line.
• Geoff Diestel asked a question:
Is there a single source containing a table illustrating all the known embeddings of classical Banach spaces into other classical Banach spaces?
It would also be nice if there was a single source with the proofs of these embeddings and counterexamples of spaces that cannot embed into other spaces. If the Handbook of the Geometry of Banach spaces ever comes out with a new edition, this would be a useful appendix.
Is L_p compact in L_{p,\infty}? More generally, is L_{p,q} compact in L_{p,r} if r>q?
Assume the underlying measure is a probability measure. I think I've heard this is true but I could be wrong.
Geoff Diestel · Texas A&M University System
Thanks Anton and good to hear from you again. Actually, the real issue is whether there are meaningful spaces in which weak convergence implies convergence in measure. I talked with Chris Lennard about this and he explained that this would be a weakened Schur property but it is unclear if there are such spaces which don't already have the schur property. With regard to your comment on compactness, I am interested in which classical spaces do and do not compactly embed into L_0. Moreover, are there any known reasons for why such embeddings would or would not exists?
• Geoff Diestel asked a question:
Which Banach spaces can be compactly embedded into L_0 and which cannot?
I am looking for a list of classical spaces which are known to compactly embed into the space of measurable functions and any which are known NOT to compactly embed into L_0. L_0 is just the space of measurable functions over the interval (0,1) equipped with Lebesgue measure.
What are the differences between "Rieman integrable" function and "Lebesgue integrable" function?
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Geoff Diestel · Texas A&M University System
I think one of the most overlooked and simple differences between the Riemann and Lebesgue integral is the fact that the preimage set function is better behaved than the image set function. The preimage commutes with unions and intersections whereas the image set function does not commute with intersections. That is f[A intersect B] is a subset of f[A] intersect f[B] but may not be equal. This is what is hiding behind the scenes that makes Lebesgue measure better.
• Geoff Diestel asked a question:
Is there a good place to begin studying compact embeddings into the space of measurable functions?
By space of measurable functions, I mean L_0(m) where m is a non-atomic sigma-finite measure space.
If H is a Hilbert space, M is a subspace of H, and A subset of H such that M \subset CL(A) - is A dense in M?
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Hamid Rezaei · Yasouj University
There exists Hilbert space operators T and non-trivial subspaces M such that M \subset cl[orb(T, x)] for some vector x while M \cap orb(T, x) is not dense in M.
Is there a way to estimate the meanvalue basing on orthogonal bases(p-adic)?
The main procedure is that: firstly we cut off finite points from the sequence. Then by applying these finite points, we can define intervals, the intervals can be fraction(denominator are added by integer); they can also be continued fraction. Then，we apply the scaled methods, firstly we should determine the value of two ratios r and r'（pay attention that : when we calcular the value of r and r' ,we can omit the repeated items）then，by using this ratios we can determine the size of meanvalue. Two crucial points we should mention is that: in this algorithm we should search another dn for comparison and we use the orthogonal base to define intervals at first.also note that the one to one correspondence between the definition and the inequality which we use to determine the meanvalue. Page 13-28
Cheng Tianren · South China Agricultural University
I have a question that whether this method can be used in number theory?
Can we use the weak fixed point property to handle the Schauder conjecture?
The Schauder Conjecture is that every compact convex subset of a metric space has a fixed point, established by Cauty in 2001. My question is: can we use the weak fixed point property to handle the Schauder Conjecture? I list the procedure in the paper: “weak fixed point property and Schauder Conjecture” and hope visitors can read it for me and answer whether this procedure is feasible? Here, I list the general ideal and the procedure: 1.use the error estimate to study the separable property. 2.to get an inequality relate the p in weak fixed point property with the bound for the projection operator 3. to get an inequality in some special dimension about the Schauder Conjecture 4.use the inequality we get in step 3 to relate the separable property (step 1) with the parameter k in weak fixed point property by characterizing the compact group 5. to get different a dimension, we get different bounding of the k in the weak fixed point property. I guess that the boundary of the parameter k in weak fixed point property supports that we have a fixed point to the metric space with compact convex subsets.
Cheng Tianren · South China Agricultural University
Thank you very much! I have read some part of cauty's proof of schauder conjecture! And my idea is different from his,i want to use the weak fixed point property. In my paper i prove that when we add the condition of the schauder conjecture to the property of the weak fixed point ,it seems that the parameter of the compact group has a bound -page 26 formula(47),whether this bound can relate the schauder conjecture and the weak fixed point?
• Cheng Tianren asked a question:
How can I calculate the included angle by studying the circle norms in convex cone?
In convex cone we always construct the circle norms, for example: x^2+y^2>1/9 x=a(1,x2,x3,...)+b(-1,x2,x3,...) our method is to analyze the closed ball about the nontrivial convex cone in a normed space and get a contradiction first,where we get a relationship between the angle and the parameter 1/k,1/n. Then we use tools in plane geometry and analytic geometry to study the circle norms and get another relationship between the included angle and the position angle.our main problem is that how to combinate step 1 and step 2,so that we can determine whether the closed balls are diviation or tangent.
Is there a classification of all subspaces of measurable functions for which weak-convergence implies convergence in measure?
I ask because this seems to allow one to replace compactness with weak-sequential-compactness(and weak-compactness for Banach spaces of measurable functions) in some arguments. For example, if X is such a space of measurable functions then any weakly-sequentially-compact subset will have the property that convergence in measure is equivalent to convergence in X.
Geoff Diestel · Texas A&M University System
Thank you!
Does the collection of all self adjoint operators have any property?
There are unitary, self adjoint and normal operators in operator theory. Do each collection posses nice property?
Azita Mayeli · City University of New York,- Queensborough
The class of self-adjoint operators have many properties. Among those are: They are compose of an operator and its adjoint. And, their spectrums are all real.
• Cheng Tianren asked a question:
Can we use the symmetry in L(3,∞) space to study the velocity ratio of the navier-stokes equation?
In the navier-stokes equation, the Ls case such that s is included in (3,∞) is studied by ladyzhenskaya. In our research, we discover a new symmetry in the Ls case that when the parameter C is equal to [(1/2)3/2]/10,the L3 space holds; otherwise when the parameter C is equal to −[(1/2)3/2]/10, the L infinite space holds; our problem is how to apply this symmetry. A practicable way we propose is that we can use the velocity ratio to construct the Ls type space. And our method is based on an inequality that the integral of the velocity ∫v2 is less that v2, eventually we get this inequality by using the hermit polynomial.
• Cheng Tianren asked a question:
What is the relation between cartesian and hilbertian?
A norm space, if its every finite dimension linear subspace has orthogonal base, we call it Cartesian. If its every one dimension subspace has orthogonal complement，that is Hilbertian. If a norm space is Hilbertian, then it is Cartesian. But, if a space is cartesian, is it Hilbertian? In the complete case, this composition is true. But in the dense case, that is uncertain. Here, I list a new relationship between cartesian and hilbertian. I wrote this relationship in the paper "some problems on orthogonal cartesian spaces" and another paper named "nonarchimedean analysis-the application of symmetric methods" and our procedure is: (1) find the growth mode of the theradius of the closed ball and use the the modulus n remainder k group to represent the theadius (page 10 to 13 in "some problems on orthogonal cartesian spaces"), (2) use the helly theorem to get the relation estimate (page 9 formular (11) in "some problems on orthogonal cartesian spaces"), (3) use an inequality to relate the group in (1) and the estimate in (2). Then we should consider the limit form of the estimate we get in (2) by applying the inequality we use in (3), which lead that the c and the k in the estimate (2) are equal when they converge to infinite in the meaningful of limit, so we can get the last estimate (refer to the example five page 20 to 23 in "nonarchimedean analysis-the application of symmetric methods") which relate cartesian and hilbertian.
What is so-called globally attractive and locally attractive?
I am a student who is now working for an project about Lyapunov stability. As I am not a mathematics student, I don't know these two notions. Could someone tell me ?
Aria Tsam · Aristotle University of Thessaloniki
What is the structure of a set of Fourier Integral operators?
There is a structure of infinite dimensional Lie group on a group of invertible Fourier integral operators of order 0. I would like to know whether one can put, eg a structure of algebra, on (possibly non invertible) Fourier integral operators, compatible with the standard structure of the Von Neuman algebra of bounded operators on a Hilbert space.
Elmar Schrohe · Leibniz Universität Hannover
You need clean intersections in order to be able to compose the canonical relations (and hence the operators), see the remarks after Def. 21.2.12. As for the symplectomorphism, one wold have to specifiy more properties.
Consider a family of real functions on metric space X. Is there any relation between separability of X and equicontiuity of this family of functions?
Equicontinuity
Pedro Terán · University of Oviedo
Can anyone help with this problem?
If B(x,1/k) is open ball in a Banach space M and there exist open set G_k in M such that G_k \subset B(x,1/k) for all natural number k. What can we say about the family {G_k} when k \to \infty? is G_k stay open ? Can we say the family converges to x?
Milen Ivanov · Sofia University "St. Kliment Ohridski"
And what is then $\limsup_k G_k$? If it is the set of all cluster points of all sequences $(x_k)$ s.t. $x_k \in G_k$, then $$\limsup_k G_k = \{x\}$$ simply because your condition implies $\|x-x_k\| \le 1/k$ for any such sequence.
What is the definition of supporting functional of the convex set C at point x0?
Compare its definition in wiki and in the book (Foundations of Mathematical Optimization Convex Analysis without Linearity) the note follows Proposition 6.1.17. - Is there a contradiction between both definitions?
Milen Ivanov · Sofia University "St. Kliment Ohridski"
Yes, but greater/lesser or equal, that is, $\ge$ or $\le$. It is not possible $f(x0)>f(x0)$ and $x0$ is in $C$.
If the image of a set is dense does it imply that the set itself is dense?
Note that the function is surjection continuous from a Banach space to a Banach space.
Milen Ivanov · Sofia University "St. Kliment Ohridski"
No! Take a projection $f$ from $X$ onto proper subspace $Y$. (Assume $X$ Hilbert if you like.) Then the image $f(Y)=Y$ of $Y$ is dense (in fact it is the whole space), while $Y$ is not dense in $X$.
What is the dimension of \cup_{i=1}^{\infinity} A_i?
Let X be infinite dimensional Banach space and A1,A2,A3,... are finite dimensional subspace of X. What about the dimension of the union of all Ai's?
Milen Ivanov · Sofia University "St. Kliment Ohridski"
May be finite or infinite depending on situation. No hard rule and/or Theorem here.
Does this approximation preserve convexity?
That is, if $f$ is convex, is its approximation convex? If yes, it can be used to solve in positive a question from the monograph of Deville, Godefroy and Zizler "Smoothness and Renormings in Banach Spaces", if an arbitrary norm on Hilbert space can be approximated by C^2 norm with arbitrary precision. (I can specify the page, but not right now, since I do not have the book near by.)
No, this approximation method does not preserve convexity. Nevertheless the problem you mention has already been solved (for arbitrary equivalent norms in a Hilbert space) by Deville-Fonf-Hajek in a Studia Math. paper from the nineties.
Does anyone have an example of a periodic function in the space $L^2(R^N)\cap L^\infty(R^N)$ ?
It will be very helpful in finding large energy solutions of some problems with lack of compactness.
Vladislav Babenko · Dnepropetrovsk National University
As far as I understand Francesco Di Plinio and A. Nandakumaran are right
• Can we say that a linear space X is not reflexive?
Suppose X be a linear space. My question is: How can we say that X is not reflexive in any norm. Reflexive means X is linearly isometric to its second dual X''
Prasanth G. Narasimha-Shenoi · Government College Chittur
Ok David, Now i got
Can someone share examples of topological vector space?
Examples with explanation will be much better
Qefsere Doko Gjonbalaj · University of Prishtina
All normed vector spaces, and therefore all Banach spaces and Hilbert spaces, are examples of topological vector spaces. A topological field is a topological vector space over each of its sub-fields. A Cartesian product of a family of topological vector spaces, when endowed with the product topology is a topological vector space.
Which condition(s) can be added to a one parameter grouop of bounded operators in such a way that its generator is a ($\sigma,\tau$)-derivation?
Counstracting a new dynamical system based on ($\sigma,\tau$)-derivation as its generator.
What are applications of Reproducing kernel Hilbert Spaces in pattern recognition?
Is there any relation between fractal interpolation function and Kernel function?
Abhishek Bhattacharya · Institute of Engineering & Management
The problem of classifying an observation into one of several different categories, or patterns, is considered. The observation consists of a sample function of a continuous-time parameter stochastic process observed over a finite-time interval. When only two categories are involved the general pattern recognition problem reduces to the signal detection problem. The methods used are based upon results from the theory of reproducing kernel Hilbert spaces. This theory has been developed within the last few years and the application of these results to problems of statistical inference for stochastic processes has taken place only recently. Therefore, a reasonably serf-contained exposition of the results required from the theory of reproducing kernel Hilbert spaces is presented. It is pointed out that the decision rule employed by the optimum pattern recognition system is based on the likelihood ratio. This quantity exists fi, and only if, the probability measures are equivalent, i.e., mutually absolutely continuous with respect to each other. In the present work only Gaussian processes are considered, in which case it is well known that the probability measures can only be either equivalent or perpendicular, i.e., mutually singular. It is shown that the reproducing kernel Hilbert space provides a natural tool for investigating the equivalence of Gaussian measures. In addition, this approach provides a convenient means for actually evaluating the likelihood ratio. The results are applied to two pattern recognition problems. The first problem involves processes which have the same covariance function but different mean-value functions and the second problem concerns processes with different covariance functions and zero mean-value functions.
The space of Riemann integrable functions
Let R[a,b] denote the space of all real (or complex) valued functions that are Riemann integrable on [a,b]. Endowed with the sup-norm (not ess sup) R[a,b] is a Banach space. What is known on the Banach space properties of R[a,b] ? It is clearly non-separable. Does it possess the predual? Is there a space X such that the dual of X is isomorphic to R[a,b]? What is R[a,b]/C[a,b]?