Banach Space - Science topic

Actually this is very subtle. Depending on the choice of quasi-linear function used to define the quasi-norm, you can get a space that embeds into Lp for all 0<p<1. Such space necessarily contains a basic sequence. However, with a different quasi-linear function, the space obtained may be minimal. For locally-bounded F-spaces this is equivalent to the lack of any basic sequences.

Condition (1): Any of its closed infinite-dimensional subspaces can contain X.

Condition (2): For a vector topology T, it is the case if and only if it can be extended on all of X that T, the only weaker Hausdorff topology on a subspace of X is a strictly weaker T.

Question 1: Is it the case that (1) is a consequence of (2)?

Answer: No in a word, and in more words, condition (1) Is not a consequence of condition (2).

Proof:

Always define a connecting plane or union of spaces, this is what induction in geometry is all about or what PT is about.

On the contrary, about direct embeddings/inclusions onto infinite dimensional closed subspaces.

A counterexample exists: let us take p dimension in space of sequences for p≠2 with ℓp⊕ℓ2 representation.

Condition (1) is false, hence, it is not a topology in exploration that can be embedded into an infinite dimension.

However, (2) satisfy as every weaker topology practically extend to any space.

Condition (2) extend to radial B-space and it is expected to be implemented because such space radially extend to any dimension.

Question 2: Let X be a reflexive space: does it at every point must be equipped with a separable Hilbert space as its centre?

Answer: Yes, a separable Hilbert space is equipped as centre, but on assumptions that X is reflexive and X satisfies both conditions (1) and (2).

Proof: The above proof clearly indicates that (i) there must be certain separation, meaning X is enough to consider super reflexive.

Indications also suggest that such properties violate containment, similar to (ii) and X must contain Radon-Nikodym property.

I'm looking for a single Banach space X that is type p for all 0<p<2 but not Hilbert. Each L_p has type p but is not type r for r>p. Also, I'm not sure the "Minkowski space" mentioned above qualifies as it does not appear to be complete but. Maybe I am misunderstanding.

Mr. @Dauphin Mwami Kamwanga Wa Kamwanga:

Mr. @Qingping Zeng is talking about the closedness of the sum of a closed left ideal and a minimal left ideal of a Banach algebra.

Which book you are looking for? If you search the name, google shows both the books.

Dear Geoff Diestel, you can benefit from this references :

1- Qingying Bu & Joe Diestel. OBSERVATIONS ABOUT THE PROJECTIVE TENSOR PRODUCT OF BANACH SPACES, I

2- Jan H. Fourie ∗,1 and Ilse M. Röntgen 2. Banach space sequences and projective tensor products

Priyanka Priyadarshini Behera

have a look at section 5.4.2 of the enclosed chapter

Best, Vladimir

Thanks

Let X and Y be Banach spaces, n a positive integer.

A function P from X into Y is a n-homogeneous polynomial if there exists a

n-linear function L from X^n to Y such that P(x)=L(x,...,x).

A 0-homogeneous polynomial is a constant function.

A polynomial is the sum of a finite number of homogeneous polynomials.

Thanks, don't know what I was thinking.

A search with keywords "weak fixed point property" (which is the official name of the property you are interested in) and with "weak normal structure" (which is a widely used sufficient condition for this property) may give you a lot of information on the subject.

Last call.If anyone interested .

Manan Anilkumar Maisuria $R^{d}$ is the d dimensional Euclidean space, you may consider n instead of d.

Hi professors. I hope you are doing well. This is a mathematical article that proves a new recurrence relation that is fundamental for mathematics. The article proves also that four infinite series are equivalent. Hence, this article opens new opportunities to demonstrate and develop new mathematical findings and observations. This is the link: https://www.researchgate.net/publication/364651911_A_useful_new_equation_of_four_infinite_series_and_sums_by_using_a_new_demonstrated_recurrence_relation

Kernel of function means collections of informations or snippes about the function,knowledge ,place and descriptions or such job depending on Google,C.V, experience .......etc

I am interested in the applications of Fixed Point Theory to differential and integrodifferential equations (fractional o generalized). Can be?

To the best of my knoeledge, the problem remains open in the more concrete setting of linear operators between Banach spaces, endowed with the usual operator norm. It may sound strange, but even the finite-dimensional case remains open! I have personally studied the problem, along with some co-authors, exclusively in the Banach space context, and mostly in the finite-dimensional case. I think that it is a particularly hard problem, even in this far more restricted scenario.

Without a shadow of doubt, the problem deserves far more attention from the functional analysis community, and therefore, I must thank you for focussing on such a simple yet deep question.

Having said all these, I do know of some interesting results by Lindenstrauss and Perles, Lima, Sharir, T. S. S. R. K. Rao among others on this topic. It might be fruitful to look into their ideas and results (again, mostly in the Banach space context). We have also tried to contribute something to this intriguing topic. We (Paul, Mal, Sain) we have obtained a complete geometric characterization of extreme contractions between two-dimensional strictly convex and smooth Banach spaces. We (Paul, Ray, Roy, Bagchi, Sain) have also been able to obtain some interesting conclusions on extreme contractions between polyhedral Banach spaces. Recently, with Paul and Sohel, we have also obtained a complete characterization of extreme contractions between finite-dimensional polyhedral Banach spaces. Most of these papers are available on RG and please feel free to look at them, should you choose to!

Of course, much remains to be done on this fascinating question, and from multiple points of view, including analytic, grometric and combinatorial. In short, a potential exciting journey into the unknown!

see

Tensor products of normed and Banach quasi *-algebras

Adamo, Maria Stella, Fragoulopoulou, Maria

Journal:

Journal of Mathematical Analysis and Applications

Year:

2020

Dinu Teodorescu

For every $f \in X^*$ and every $a \in X$ consider the following $F: X \to R$: $F(x) = f(x) + (f(x-a))^2$. The derivative of $F$ at point $a$ is equal to $f$.

This answers your question "Every element of continuous dual is a "differential of a nonlinear function"? It's generally true? Why?".

As I remember, if we have a topological vector space (X,T) - T the linear topology on X, then a linear topology U on X is Mackey topology iff all linear and continuous functionals on X relatively to T are also continuous relatively to U. In other words, a Mackey topology is the maximal topology on X, preserving the continuous dual of (X,T). And, if (X,T) is locally convex( T is generated by the family of Minkowski functionals on X), then T is right Mackey topology on X.

Please correct me, if I'm wrong!

Now let's consider the Lorentz space L^1,q. If 1/q+1/r=1 and L^1,r may be identified with the dual of L^1,q, I think this fact would help to obtain an answer to your question. But I don't know anything about duals of Lorentz spaces and my assertion is probably not true.

Thank you very much.

I have found an answer in the book "Applied Nonlinear Analysis" by Ekeland and Aubin, and also in the Notes by Ralph Howard: "The Inverse Function Theorem for Lipschitz Maps" (inverse.pdf)

Dear Nima Saliani

Norms kill imaginary parts.

So where is your problem?

Regards

I suggest you the 2-uniformly convex hyperbolic spaces.

Six-dimensional space is any space that has six dimensions, six degrees of freedom, and that needs six pieces of data, or coordinates, to specify a location in this space. There are an infinite number of these, but those of most interest are simpler ones that model some aspect of the environment. Of particular interest is six-dimensional Euclidean space, in which 6-polytopes and the 5-sphere are constructed. Six-dimensional elliptical space and hyperbolic spaces are also studied, with constant positive and negative curvature.

Formally, six-dimensional Euclidean space, ℝ6, is generated by considering all real 6-tuples as 6-vectors in this space. As such it has the properties of all Euclidean spaces, so it is linear, has a metric and a full set of vector operations. In particular the dot product between two 6-vectors is readily defined and can be used to calculate the metric. 6 × 6 matrices can be used to describe transformations such as rotations that keep the origin fixed.

More generally, any space that can be described locally with six coordinates, not necessarily Euclidean ones, is six-dimensional. One example is the surface of the 6-sphere, S6. This is the set of all points in seven-dimensional space (Euclidean) ℝ7 that are a fixed distance from the origin. This constraint reduces the number of coordinates needed to describe a point on the 6-sphere by one, so it has six dimensions. Such non-Euclidean spaces are far more common than Euclidean spaces, and in six dimensions they have far more applications.

Every Banach lattice with order continuous norm has the BD property, but it does not have the weak GP propetry, in general.

Dear Filip Soudský

On every infinite-dimensional linear space (real or complex) there exist non-equivalent norms. Usually this statement is given as an exercise, like for example Exercise 4 in Section 10.2.1 of my book ``A Course in Functional Analysis and Measure Theory'' https://www.springer.com/us/book/9783319920030#aboutBook

Let me give here a possible solution. Let $X$ be an infinite-dimensional linear space. Select a Hamel basis $e_t, t \in T$ for $X$ (see the definition and the existence theorem in sections 5.1.1 and 5.1.3 of the same book). Every element $x \in X$ is uniquely representable as a linear combination of $e_t, t \in T$, that is in the form

$$

\sum_{t \in T} a_t(x) e_t,

$$

where all $a_t(x)$ are scalars and only finite number of $a_t(x)$ are different from zero.

Define two norms: $||x|| = \sum_{t \in T} |a_t(x)|$ and

$|||x||| = \max_{t \in T} |a_t(x)|$. Both norms are correctly defined, because in the corresponding sum and max only finitely many nonzero coordinates are involved,

but these two norms are not equivalent.

Best, Vladimir Kadets

Dear Luisa Malaguti

In general, the answer is negative by the following indirect argument. Assume X is separable and uniformly convex, and let G: X/K -->X be the map in question. If this map is Lipschitz, then it is a Lipschitz embedding, because the inverse map G^{-1} : G(X) \to X/K is also Lipschitz (in fact, it is a contraction) . This means, that X/K is Lipschitz embeddable into X. According to Theorem 3.5 of the paper

Heinrich, Stefan; Mankiewicz, P. Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces. Stud. Math. 73, 225-251 (1982),

this implies that X/K is isomorphic to a subspace of X. But the general result "each quotient space of a uniformly convex separable Banach space X is isomorphic to a subspace of X" is not correct. Say, L_5[0,1] has a quotient space isomorphic to l_4, but does not have subspaces isomorphic l_4.

This latter fact can be extracted from two old papers:

Kadets, M. I. Linear dimension of the spaces \(L_{p}\) and \(l_{q}\).

Usp. Mat. Nauk 13, No. 6(84), 95-98 (1958)

Kadets, M. I.; Pełczyński, Aleksander

Bases, lacunary sequences and complemented subspaces in the spaces \(L_ p\).

Stud. Math. 21, 161-176 (1962).

almost surely, the results are contained in some books like Lindenstrauss-Tzafriri Classical Banach spaces I and II.

All the best, Vladimir

Dear Miodrag,

Fist here is definition of the Henstock–Kurzweil integral :

The set of KH-integrable functions forms an ordered vector space and the integral is a positive linear form on this space. On a segment, any Riemann-integrable function is KH-integrable (and of the same integral).

For relation between the gauge integral and the Lebesgue and Riemann integrals, i suggest you to see links and attached file on topic.

https://arxiv.org/pdf/1609.05454.pdf

https://math.vanderbilt.edu/schectex/ccc/gauge/

Best regards

Yes, we do have to know about the spaces, at least during our university studies, to enables us to expand our mind into abstact level, that will be very usefull for design activities

The answer is negative. Take as $X$ the space $\ell_1$ in the LUR renorming $\|x\| = \sqrt{\|x\|_1^2 + \|x\|_2^2}$ where $ \|x\|_p$ denotes the canonical $\ell_p$ norm. This $X$ possesses quotient spaces that are not strictly convex. With some additional effort one can demonstrate that $X$ has a quotient isometric to $\ell_1$ in canonical norm, which implies that $X$ has as its quotients all separable Banach spaces.

If Y is dense in X, then any linear continuous form on Y has a unique linear continuous extension to the whole space X. Hence, in this case, Y* is contained in X*. On the other hand, the restriction to Y of any linear continuous form on X is linear and continuous on Y, for any vector subspace Y of X, even in the case when Y is not dense in X. Thus, it seems that always X* is contained in Y*.

In general, it is a difficult problem to identify the points at which an operator attains norm, if it attains norm at all! Having said that, there are certain estimations available in l_p^2 spaces over the real field, where p \neq 1, 2, \infty. It says that if an operator attains norm at more than 2(8p-5) number of points, then it must be a scalar multiple of an isometry.

For p=2, norm attaining point is unique (upto multiplication by \pm 1) if and only if the concerned operator is a smooth point in the operator space. In general, for a SMOOTH operator between Banach spaces, norm attaining point is unique (upto multiplication by \pm 1) if the operator attains norm.

Attached photos of my definition of hypercyclic infinity tuples.

As you point out the Hermite functions are the energy eigenfunctions for the harmonic oscillator. One aspect here is that the Schrodinger equation for the harmonic oscillator is invariant under the Fourier transform (the second derivatives and x^2 transform into each other).

One way to think about these functions is that they are the result of doing the Graham -Schmidt orthonormalization process starting with the functions 1,x,x^2,x^3, ... The polynomials are dense in the functions that are square integrable with respect to e^{-x^2}dx and so you get an o.n. basis from this process.

If you take *any* sequence of functions whose linear combinations are dense, you can do this same process to get a different o.n. basis. You can even do the polynomials in a different order. Depending on the permutation of the integers used, you may get *very* different results.

For example, do x^0, x^1x^,3, x^2, x^4, x^6, x^5, x^7,x^9, x^11,..

where you take one even exponent, two odd, three even, four odd, etc.

There is a long story of dependence of the Baire's theorem on AC (Axiom of choice). In fact, it comes from the work of P. Bernays in 1942. The Baire's theorem depends on a weaker form of AC, called Axiom of dependent choice (AD).

I am recommending a short Wikipedia article: https://en.wikipedia.org/wiki/Axiom_of_dependent_choice

To study these issues more in-depth, start with the literature at the end of this article. About 99% of the mathematicians who use the Baire's theorem, Banach open mapping theorem, or Banach-Steinhaus theorem, really do not care whether the above depend on AC or DC. Nevertheless, connections with the foundations of mathematics are very important here.

In Mirotin's answer one of the spaces is not complete, but the construction can be easily modified to obtain an operator that acts in one the same Banach space $X$.

Namely, in the infinite-dimensional Banach space $X$ choose a linearly independent sequence of vectors $\{{e_n}\}_{n = 1}^\infty$, and choose a set $A \subset X$, such that $A\cup \{{e_n}\}_{n = 1}^\infty$ is a Hamel basis of $X$. Now define the operator $ T\colon X \to X $ by the following rule: for $x\in A $ put $ Tx = x$, put $Te_1 = 0$ and for the vectors $\{{e_n}\}_{n = 2}^\infty$ put $ Te_n = n e_{n-1}$, and then extend $T$ to the whole space $X$ by linearity.

This $T$ is surjective, because each because each element of the Hamel basis $A\cup \{{e_n}\}_{n = 1}^\infty$ belongs to $T(X)$; $T$ is not injective, because $Te_1 = 0$, and is unbounded because $ Te_n = n e_{n-1}$ for $n = 2, 3, ...$.

The definition of Hamel basis you can find in section 5.1.3 of my book https://link.springer.com/book/10.1007%2F978-3-319-92004-7

Many facts and examples related to your question you can find in Chapter 10 of the same book.

The question you ask is a very broad one. Since differential operators are unbounded - the definition of the on which one defines the operator is important. When one goes to infinite dimensions, there might not even be a point spectrum (eigenvalues) - the spectrum may be only the continuous spectrum and residual spectrum. If the operator has a self joint extension or a normal extension then the problem becomes more tractable but for non-normal operators the problem is quite complex.

At one time a class of operators "spectral operators" were defined, see Dunford and Schwartz, "Linear Operators", vol III, to address the issues on non-normal operators. Proving a non-normal operator - even an ordinary differential operator - was not a trivial task. The goal of this effort was to expand the concept of Jordan Canonical form to first bounded linear operators on a Hilbert/Banach space and then to unbounded linear operators, e.g., differential operators. For compact operators or operators with compact resolvents - that has been solved. For bounded normal and unitary operators, that has been solved (spectral theorem). For unbounded operators with a compact resolvent, that has been pretty much solved (by using the spectral theorem the spectral theorem). For unbounded self adjoint operators, there is also a spectral theorem. However, for non-normal bounded operators and non-self adjoint unbounded operators - there is no general result on a canonical form.

In these problems stability questions be quite problem specific.

Dear Abeer,

I happen to share the same field of interest as yours. You may have a look at my ResearchGate publications, and more importantly, the references therein, to find some recent publications on orthogonality in Banach spaces.

I work mostly with Birkhoff-James orthogonality only, but there are excellent works on other kind of orthogonality types in Banach spaces, e.g., isosceles orthogonality.

It might also be a good idea to begin with the three famous papers by R. C. James on Birkhoff-James orthogonality in normed linear spaces. You will find all these references, and more, in our work, and recent works by several other mathematicians.

First, recall that if $X$ is a normed space, $X_1$ and $X_2$ be linear subspaces of $X$, and $X = X_1 \oplus X_2$, then the natural projector $P$ onto $X_1$ parallel to $X_2$ is defined as follows:

given $x\in X$, decompose it as $x = x_1 + x_2$, where $x_1 \in X_1$, $x_2 \in X_2$ (by assumption, this decomposition exists and is unique), and then put $P(x) = x_1$.

This linear projector $P$ is continuous if and only if there is a constant $C > 0$ such that for every $x_1 \in X_1$, $x_2 \in X_2$ the inequality $\|x_1\| \le C \|x_1 + x_2\|$ holds true.

The most natural way of constructing examples of closed subspaces with non-closed sum is based on the following theorem (which is a standard application of the closed graph theorem):

Let $X$ be a Banach space, $X_1$ and $X_2$ be closed subspaces of $X$, and $X = X_1 \oplus X_2$. Then the natural projector $P$ onto $X_1$ parallel to $X_2$ is a continuous operator.

Consequently, if in a Banach space $E$ you construct two closed subspaces $X_1$ and $X_2$ with zero intersection, denote $X = X_1 \oplus X_2$ and find that the projector $P$ onto $X_1$ parallel to $X_2$ is discontinuous, then $X$ must be non-closed (otherwise $X$ would be a Banach space and by the previous theorem $P$ should be continuous).

There are many such examples of $X_1$ and $X_2$. For example, if in the Hilbert space $\ell_2$ you take an orthonormal basis $e_n$, $n = 1, 2, ...$, and take as $X_1$ the closed linear hull of all $e_{2k}$, $k = 1, 2, ..$ and as $X_2$ the closed linear hull of vectors $e_{2k} + 2^{- k} e_{2k-1}$, $k = 1, 2, ..$, then these $X_1$ and $X_2$ will be the desired example.

A comprehensive study on this topic is Dunford Schwartz, Part I, p. 489-511, see also L. Meziani, Acta Math. Univ. Comenianae 74 (2005), 59-70, available at

[access May 2018]

Something is missing in the statement of the question. In the way how it is formulated now

"Let X be a Banach space which has no infinite(-dimensional) reflexive subspaces. Does this assumption imply that the space X itself is not reflexive?"

the answer is trivial: If X is finite-dimensional, then it has no infinite-dimensional reflexive subspaces, but X itself is reflexive.

If one reformulates the question adding that X itself is infinite-dimensional, then X also is a subspace of X, so by the assumption of the question, X cannot be reflexive.

Finally, if one asks the question "Let X be an infinite-dimensional Banach space which has no PROPER infinite-dimensional reflexive subspaces. Does this assumption imply that the space X itself is not reflexive",

then the answer really needs the theorem that every closed subspace of a reflexive Banach space is reflexive, which can be found in most functional analysis textbooks, for example in those which were recommended in the answer by Oleg Reinov.

Dear Professor,

Kindly go through it.

www.springer.com/cda/content/document/cda.../9780817683276-c1.pdf?SGWID..

Frechet derivative is a generalization of the ordinary derivative and the first Frechet derivative is Linear operator. When you study differential calculus in Banach spaces you need to study Frechet and Gateaux derivatives. The different between Frechet and Gateaux derivatives is that the Frechet derivative is generalization of partial derivative for multivariable functions and Gateaux derivative is generalization of directional derivative. I think that the Frechet derivative is used in the peridynamic state based model to linearized the model.

Dear Mieczyslaw,

It is well known that the weak topology is Hausdorff. Also this topology is regular. Moreover, if $\mathcal{U}=\{U_s:s\in I\}$ is an open covering of a reflexive Banach space $X$, then for every $n\in N$,

$B[0,n]=\{x\in X:\|x\|\leq n\}\subseteq \cup U_s$.

By compactness of $B[0,n]$ in the weak topology, it follows that for every $n\in N$, there exist $\alpha^n_1,\dots,\alpha^n_{s_n}\in I$ such that

$B[0,n]\subseteq\cup\{ U_{\alpha^n_i}:i=1,\dots,s_n\}$.

Thus $\underset{n\in\mathbb{N}}{\cup}\{U_{\alpha^n_i}:1=1,\dots,s_n\}$ is a countable subcovering of $\mathcal{U}$.

Therefore $X$ is Lindelöf.

Finally, since every regular Lindelöf space is paracompact, it follows that if $X$ is a reflexive Banach space then $X$ with the weak topololy is paracompact.

This is a very interesting question.

Followers of this thread may find the following papers interesting:

A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to Linear Dynamics

On (mk)-hypercyclicity criterion

Hypercyclic and mixing operator semigroups

For non-separable F spaces, see

Subseries convergence in topological groups and vector spaces

On the Orlicz-Pettis property in nonlocally convex $F$-spaces

Here is a description of a book that appears to give you what you need.

/Michael

Ordered Banach Spaces

Even though there are many excellent monographs on Banach spaces, the topic of ordered Banach spaces is not very often included. And even if it is, the authors usually deal only with the more particular case of Banach lattices. The aim of this book is to fill (at least partially) this gap and to provide an up-to-date study of the structure of convex cones contained in Banach spaces and some of the related operators. The book originates from three different sources. The first is conical measures (introduced by G. Choquet and used for developing integral representation theory). The second is the well-established theory of Banach lattices and the third is the Krivine theorem on finite representability of finite dimensional ℓp spaces in Banach lattices. By blending them together, the author obtains an original and instructive contribution to classical functional analysis.

The book starts by presenting various forms of the Hahn-Banach theorem and proving theorems of M. G. Krein, V. L. Klee and T. Ando on cones in Banach spaces. The second chapter is devoted to a presentation of B. Maurey’s theorem on factorization of operators through Lp spaces. An important ingredient here and later on is the Rosenthal lemma. To study convex cones, chapter 3 introduces conical measures and their basic properties. The next part contains the first main results of the book on p-summing operators and their factorization. The author then proceeds with an investigation of representability of finite ℓp spaces in normal cones of Banach spaces. He also shows a relationship between type and cotype of a Banach space with the index introduced in the previous chapter. The author then studies positive operators starting in C(K) spaces and he continues the investigation of p-summing operators on convex cones. The Pietsch inequality and a composition theorem are proved. The last chapter describes the situation of tensor products and positive maps. Five appendices and several open problems complete the book making the presentation rather self-contained.

Author: R. Becker

Publisher: Hermann

Published:

2008

ISBN:

978-27056-6721-4

Price:

EUR 43

it seems that elegant yet simple revolutionary paper by M.A Robdera "unified approach to vector integration "  internet journal of operator thgeory functional analysis and applications vol 5, number 2, 2013 pp119-139 pushpa publications , allahabad india contains key to the answers to my questionset

Dear Geoff,

In L_p[0,1] with p < 1 there is no continuous non-zero linear functional, consequently there is no complemented 1-dimensional subspace.

Best regards, Vladimir

One of the predual space of Morrey space with variable exponents is the block space with variable exponent, see 

Cheung K.,and Ho, K.-P. Boundedness of Hardy-Littlewood maximal operator on block spaces with variable exponent, Czechoslovak Math. J. 64(139), 155-171 (2014).

In the late 1960's Browder and Petryshyn - constructed a degree theory for a class of mappings between two different Banach spaces.

F. E, Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree  for nonlinear mappings in Banach spaces, J. Funct. Anal. 3 (1969)

Also in the late 1960's Elworthy and Tromba developed a degree theory for some types of nonlinear proper Fredholm operators (C^1 mapping whose differential at each point is linear Fredholm operator) of index zero between Banach manifolds.

K. D, Elworthy and A. J. Tromba, Differential structures and Fredholm maps on Banach manifolds, Global Analysis, vol 14-16, Proc. Sympos. Pure Math, American Math Soc., 1970.

A pretty good overview of the history of index theory can be found in 

J. Mawhin, Lerary-Schauder Degree: A half century of extensions and applications, Topological Methods in Norlin. Anal, (14), 1999.

No; consider the norm (or its square) on an infinite dimensional Hilbert space.

Dirk.

Dear Khalid,

I believe that the correct statement of the question comes from terminology of interpolation theory. See, for example

A Banach space X is said to be continuously embedded in a Hausdorff topological vector space Z when X is a linear subspace of Z such that the inclusion map from X into Z is continuous. A compatible couple (X, Y) of Banach spaces consists of two Banach spaces X and Y that are continuously embedded in the same Hausdorff topological vector space Z. The embedding in a linear space Z allows to consider the space X∩Y equipped with the norm ||z|| = max{||z||__X,||z||__Y}. In this terminology the Khalid's question means: is it true that every continuous linear functional on  X∩Y is of the form x*+y*, where  x* is a restriction to  X∩Y of an element of X*, and y* is a restriction to X∩Y of an element of Y*.

Khalid, did you mean this question?

All the best, Vladimir

Hellow everybody,

What is meant by relative interior ? If you mean the interior with respect to its own topology, it is of course non empty since any topological space is open in itself. If you mean the interior of a subset  A of a subspace F of a Banach space E with respect to the relative topology on F, then Thomas Korimort has given the answer.

Remember that : in any topological vector space, the only subspace which has a non empty interior is E itself, because the 0-neighbourhoods are absorbant..

Also a subset of a topological space can fail to be neither open nor closed.

Dear Prof. Peters, thank you. I will take a look.

All about the determinants - see

"I. Gohberg, S. Goldberg, N. Krupnik: Traces and Determinants of Linear Operators, Birkhauser Verlag, Basel-Boston-Berlin (2000)."

For commutative Banach algebras the Shilov Idempotent Theorem is definitive. Taking the unital case for simplicity, there is a 1-1 correspondence between the clopen subsets of the character space and the idempotents in the commutative unital Banach algebra.

1. What is necessary to consider and what is not depends on the circle of problems that one studies.

2. When one speaks about Banach space valued functions there is no sense in avoiding concepts that depend on the axiom of choice, because in the Banach space theory everything is based on the Hahn-Banach theorem, which depends on the axiom of choice. So without this axiom you can do nothing. Moreover, on the axiom of choice depends the existence of non-measurable functions, the existence of measurable ones does not depend. So what is the problem?

3. The more tools you know to use, the more results you can obtain.  There are several different (not equivalent) concepts of measurability for Banach space X valued function f. Scalar measurability (composition of f with any bounded linear functional x* is measurable), usual measurability (preimages of Borel sets are measurable) and strong measurability (approximability a.e. by a sequence of simple functions) are the most important ones. The interplay between these types of measurability is an important tool in Banach space theory. For example it is used in the proof of the fact that the dual space to a separable Banach space has the Radon-Nikodym property if and only if this dual space is separable.

4. An advantage of measurability is that it is preserved by many operations that do not preserve integrability. Say, you may manipulate with functions and don't care if the Bochner integrability is preserved on each step: you just need to check measurability at each step (which is usually evident)  and if what you get finally has an integrable majorant, then it is integrable. So breaking  integrability in two parts (strong measurability and existence of an integrable majorant) is very useful. 

No, because it is partial sum is not convex in general.  

If E is a Banach space, E* is its dual space (i.e. elements of E* are linear functionals on E), then there is a standard notation for action of functional f \in E* on element x from E: <f, x> instead of f(x). In this notation the definition of adjoint operator T* to operator T: X --> Y is the following: T* acts from Y* to X* and satisfies condition

<T*f, x> = <f, Tx>, or what is the same, (T*f)x = f(Tx).

B(H) is not a WCG space, so it does not fit into Plichko´s theorem. But B(H) is a dual space, because there is a locally convex Hausdorff topology \tau on B(H) that is weaker than the norm topology, and such that the unit ball of B(H) is \tau-compact.

This topology \tau is the topology of weak pointwise convergence.

Best regards, Vladimir

There is an idea to consider instead of inner product <Tx,x> the expression of the form x*(Tx), where x* is a norm-1 linear functional, x is a norm-1 element and x*(x) = 1. With this idea in hand one can generalize concepts of self-adjoint operators and positive operators. Of course, this generalization does not work as good as in Hilbert spaces, but nevertheless sometimes is useful. See the following book, where this idea is used to define self-adjoint and positive elements in Banach algebras.

F.F.Bonsall and J.Duncan,

Numerical Ranges of Operators on Normed Spaces and of Elements

of Normed Algebras, London Math. Soc. Lecture Note Series

2, Cambridge, 1971

It has been proved that the range of A+B, Note that is coercive, if X is reflexive, then range of A+B is X^*. if X is a general Banach space, we want to know, whose range whether is a linear subspace. If yes, please give a proof. If not, please give a counterexample.