All Questions
1,222 questions
4
votes
1
answer
574
views
Criterion for weak convergence of sequences
Let $E$ be a normed space and let $F\subset E^{*}$. It is known that $F$ is dense if and only if the restriction of $\sigma(E,F)$ on $B_E$ coincides with the weak topology.
Hence, if $F$ is dense and ...
- 5,529
2
votes
1
answer
799
views
Weak-* convergence in $L^\infty((0,T)\times\Omega)$ implies weak-* convergence in $L^\infty(\Omega)$ for a.e. $t \in (0,T)$?
Let $\Omega$ be a bounded and smooth domain. Suppose I have a sequence of non-negative functions $u_n \in L^\infty((0,1)\times \Omega) \cap L^\infty((0,1);L^\infty(\Omega))$ with
$$0 \leq u_n \leq 1 \...
- 73
0
votes
0
answers
161
views
When does a positive operator preserve invertibility
Let $\Omega_1,\Omega_2$ be compact Hausdorff spaces and let $P:C(\Omega_1)\longrightarrow C(\Omega_2)$ be a unital positive operator. I wanted to know if there is a necessary and sufficient condition ...
- 175
3
votes
1
answer
176
views
Sufficient condition for asymptotic-$\ell_{p}$ in terms of spreading models?
Let $(X,\|\cdot\|)$ be a Banach space with a Schauder basis and fix $p\in[1,\infty]$. Suppose that $X$ is asymptotic-$\ell_{p}$ with respect to this basis. It is known that the closed linear span of ...
- 201
3
votes
0
answers
138
views
Property $(V_1)$ for Banach spaces
This aim of this note is to record a problem that still seems to be open.
Räbiger, in his doctoral thesis, defined property $(V_1)$ as follows: A Banach space $X$ has property $(V_1)$ if every ...
- 2,605
4
votes
1
answer
171
views
Concrete example of non-nuclear operator $E \to F$ and isometry $F \hookrightarrow G$ so that the composition $E \to F \hookrightarrow G$ is nuclear
DISCLAIMER: I posted the same question a week ago on Mathematics Stack Exchange.
We know by an abstract argument that there exist Banach spaces $E$, $F$, $G$ and maps $E \to F \hookrightarrow G$ such ...
0
votes
0
answers
129
views
Certain decompositions of decomposable Banach spaces
Let $\mathcal{X}$ be a decomposable Banach space (i.e. a topological direct sum of infinite-dimensional subspaces, say $\mathcal{X}=\mathcal{A}\oplus\mathcal{B}$). Can one always obtain another ...
- 1,453
8
votes
1
answer
262
views
On $C(K)$ spaces embeddable into the Banach space $c_0$
Problem 1. Characterize compact Hausdorff spaces $K$ for which the Banach space $C(K)$ of continuous real-valued functions embeds into the Banach space $c_0$.
Since $c_0$ has separable dual, such $K$ ...
- 41.9k
8
votes
1
answer
162
views
closed ideals in L(L_1)
Denote $L_1=L_1[0,1]$ The lattice of closed ideals in $\mathcal{L}(L_1)$ includes the chain
$$
\{0\}\subsetneq\mathcal{K}(L_1)\subsetneq\mathcal{FS}(L_1)
\subsetneq\mathcal{J}_{\ell_1}(L_1)\subsetneq\...
- 1,591
9
votes
1
answer
977
views
Completed tensor product is exact
In the beginning of the 7th chapter of the book "Spectral theory and analytic geometry over non-Archimedean fields" by Vladimir Berkovich one can find the phrase "...tensor product functor is exact on ...
- 332
2
votes
1
answer
143
views
How to characterize the order convergence in Bochner-integrable functions space?
Let $(\Omega,\Sigma,\mu)$ a finite measure space. We want to characterize the order convergence (for sequences) in Bochner integrable functions space $L^1(\mu,X)$, $X$ Banach lattice.
In $L^p$ we have:...
- 21
11
votes
2
answers
513
views
What is the structure of a Banach space $X$ when $Y$ and $X/Y$ are hereditarily indecomposable?
Assume that $X$ is a separable Banach space and $Y$ a closed subspace such that
$Y$ and $X/Y$ are hereditarily indecomposable (HI). The general question is what is the possible structure of $X$.
...
- 986
6
votes
0
answers
132
views
Mazur-Ulam bases in finite-dimensional Banach spaces
Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere ...
- 4,535
3
votes
1
answer
148
views
Measurability of superposition operator with non-separable Banach space
Let $f\colon I \times X \to \mathbb{R}$ be a map where $I \subset \mathbb{R}$ is an interval, $X$ is a Banach space (possibly non-separable) and we have
$$t \mapsto f(t,x) \text{ is measurable}$$
$$x \...
- 107
13
votes
0
answers
395
views
Converse to Riesz-Thorin Theorem
Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.
The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....
- 854
5
votes
1
answer
153
views
Why is density and separability needed for uniqueness of weak (time) derivatives?
Let $X,Y$ be Banach spaces with $X \subset Y$. Recall that $u \in L^1(0,T;X)$ has weak derivative $g \in L^1(0,T;Y)$ if
$$\int_0^T u(t)\phi'(t) = -\int_0^T g(t)\phi(t) \qquad\forall \phi \in C_c^\...
2
votes
1
answer
298
views
complemented subspace of the direct sum of two Banach spaces
When I was reading a paper, I saw something like:
If $F$ and $E$ are Banach spaces with symmetric bases (precisely, they are symmetric sequence spaces), and $F$ is isomorphic to a complemented ...
- 617
7
votes
1
answer
122
views
Containment of $c_0$ in projective tensor products
Let $X$ and $Y$ be Banach spaces and denote by $X\hat{\otimes}_\pi Y$ the projective tensor product.
Question:
If $X\hat{\otimes}_\pi Y$ contains an isomorphic copy of $c_0$, must then $X$ or $Y$ ...
- 125
1
vote
1
answer
184
views
Example when Kantorovich condition would not hold
Let $K \in M_+(R_+^2), f \in M_+(R_+)$. Consider operator
$$
(T_k)(x)=\int_{R_+}K(x,y)f(y)dy, \quad y\in R_+.
$$
Denote by $f^*(t)=\inf\{\lambda>0: \alpha x \in R_+: \mu_f(y)>\lambda\}$ the non-...
- 343
3
votes
1
answer
128
views
The weak*-convergence of the summing basis of $c_{0}$
Suppose that $(x_{n})_{n}$ is a sequence in a Banach space $X$. We let $\textrm{clust}_{X^{**}}((x_{n})_{n})$ be collection of all the weak*-limit points of $(x_{n})_{n}$ in $X^{**}$.
Let $(e_{n})_{n}$...
- 3,343
3
votes
1
answer
220
views
An improvement of Sobczyk's Theorem
Sobczyk's theorem states that if a separable Banach space $X$ contains a subspace isometric to $c_{0}$, then $X$ contains a subspace $Z$ which is isometric to $c_{0}$ and is $2$-complemented in $X$. ...
- 3,343
3
votes
1
answer
561
views
Construction of an infinitely Fréchet differentiable function with given set of zeros in a Banach space
After looking at this question, I am now wondering if the following is true.
Let $X$ be a separable Banach space over $\mathbb R$ or $\mathbb C$, and $A\subseteq X$ a closed set. Then there exists ...
- 1,755
8
votes
2
answers
488
views
If the cardinality of $B(X)$, the space of operators on $X$, is continuum, must $X$ be separable?
Does there exits any non-separable Banach space $X$ such that the size (cardinal number) of $B(X)$, bdd linear operators on $X$, is just of the continuum?
- 4,058
5
votes
1
answer
224
views
reference request: unbounded operators on normed spaces
I'm looking for books where the theory (basic properties, adjoints etc.) of unbounded linear operators between locally convex spaces or at least Banach spaces is developed. In Brezis' functional ...
- 105
1
vote
1
answer
164
views
Asymptotic models and passing to sub-arrays
If, for all $k\in\mathbb{N}$, $(x_{i}^{k})_{i=1}^{\infty}\in X^{\mathbb{N}}$ is normalized and $M$-basic and if, in addition, for all $k\leq i_{1}<i_{2}<\ldots$ the diagonal sequence $(x_{i_{k} }...
- 201
0
votes
0
answers
291
views
Operator norm on tensor product of trace classes is multiplicative
Given Hilbert spaces $\mathcal H_1,\mathcal H_2,\mathcal K_1,\mathcal K_2$ and bounded linear maps $S_i:\mathcal B^1(\mathcal H_i)\to\mathcal B^1(\mathcal K_i)$, $i=1,2$ between the respective trace ...
7
votes
4
answers
1k
views
Radon-Nikodým property of $\ell^\infty$
I am wondering whether $\ell^\infty(\mathbb N)$ has the Radon-Nikodým property. Of course $\ell^1(\mathbb N)$ does, but I was unable to find out whether (e.g.) duals of spaces with the R-N property ...
- 3,388
4
votes
2
answers
1k
views
Characterizations of Wiener algebra
The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that
$$
\mathcal W\subset ...
- 16.2k
9
votes
1
answer
1k
views
Luxemburg norm as argument of Young's function: $\Phi\left(\lVert f \rVert_{L^{\Phi}}\right)$
Let $\Phi$ be a Youngs's function, i.e.
$$ \Phi(t) = \int_0^t \varphi(s) \,\mathrm d s$$
for some $\varphi$ satifying
$\varphi:[0,\infty)\to[0,\infty]$ is increasing
$\varphi$ is lower semi ...
- 193
3
votes
1
answer
788
views
Higher order functional derivatives
Let $E, F$ be Banach spaces. A continuous bilinear functional ${\langle \cdot\,, \cdot \rangle }: E \times F \to \mathbb{R}$ is called $E$-non-degenerate if $\langle x,y\rangle = 0$ for all $y \in F$ ...
- 3,525
2
votes
0
answers
347
views
Can quotient space be isomorphically isometric to some closed subspace of original one?
Suppose $\mathcal{B}$ is a Banach space which is not isomorphic to a Hilbert space. What condition should I pose on $\mathcal{B}$ such that for every closed subspace $\mathcal{M}$, the quotient space $...
- 53
3
votes
1
answer
260
views
Reference request: completion of Banach norm on sum
Let $X_1,X_2$ be Banach subspaces of a locally convex space $X$. Then the subset
$$
X_1+X_2 = \left\{
x\in X:\, x= \beta_1 x_1 + \beta_2 x_2 \, \beta_i \in \mathbb{R},\, x_i \in X_i
\right\},
$$
a is ...
- 5,405
4
votes
1
answer
344
views
Ideal of strictly singular operators
Let $X$ be a Banach space. An operator $T:X\to X$ is called strictly singular iff for any infinite dimensional subspace $Y\subseteq X,$ $T|_{Y}:Y\to T(Y)$ is not an isomorphism.
It is known that for $...
- 1,879
3
votes
0
answers
246
views
Regularity of the dependence of the flow on the vector field definining it
Let $M$ be a smooth compact manifold and $k \geqslant 1$.
Define $\mathfrak{X}^k(M,TM)$ to be the set of vector fields $M \rightarrow TM$ of class $C^k$. As $M$ is compact, endowing $\mathfrak{X}^k(M,...
3
votes
1
answer
178
views
Banach Mazur distance between the cube and the cross-polytope in the dimensions for which a Hadamard matrix exists
The Banach-Mazur distance between two centrally symmetric convex bodies $K,L\in\mathbb{R}^n$ can be defined as
$$ d(K,L) = \inf \{ r : \exists T\colon \mathbb{R}^n \to \mathbb{R}^n \text{ linear such ...
- 183
5
votes
1
answer
177
views
An extremal property of points on the unit sphere of a 2-dimensional Banach space
Let $(X,\|\cdot\|)$ be a 2-dimensional real Banach space and $S=\{x\in X:\|x\|=1\}$ be its unit sphere. Assume that $S$ is smooth in the sense that for any $y\in S$ there exists a unique functional $y^...
- 41.9k
4
votes
1
answer
472
views
Is the set of weak*-continuous operators closed in the weak*-operator topology?
I recently came across this unanswered MO question an answer to which I would also be interested in. However the formulation of said question is somewhat imprecise and lacking detail in my opinion so ...
4
votes
0
answers
145
views
Hamel basis with all coordinate functionals discontinuous
If $X$ is an infinite dimensional (separable) Banach space, can one find a Hamel basis $(x_\alpha)_{\alpha\in\Lambda}$ such that all coordinate functionals $x_\alpha^*(x_\beta)=\delta_{\alpha,\beta}$ ...
- 1,361
2
votes
0
answers
89
views
Another question about asymptotic models in Banach spaces
The array $(x_{i}^{k})_{i=1,k\in\mathbb{N}}^{\infty}$ of normalized $M$-basic sequences in a Banach space $X$ is itself called $M$-basic if, for every $k\leq i_{1}<i_{2}<\ldots$, the diagonal ...
- 201
3
votes
1
answer
170
views
Integration on quasi-Banach spaces and Schatten ideals
Let $[a,b]$ be an interval and $X$ a Banach space (for starters). We know that continuous functions $f:[a,b]\to X$ are Riemann integrable. Suppose now that $X$ is a quasi-Banach space, that is, its ...
- 143
2
votes
2
answers
156
views
Can we say that : $ (A-B)\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $
Let $X$ be a separable Banach space and $A,B$ are closed convex subsets of $X$ such that $B\subset A$ and
$$
A\cap\overline{B}(0,r) \text{ and } B\cap\overline{B}(0,r) \text{ are weakly compact, } \...
- 117
4
votes
0
answers
2k
views
Eigenvalues and spectrum of the adjoint
In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$.
But in infinite dimensions this need no longer be ...
- 3,873
13
votes
2
answers
653
views
The geometry of $\mathbb{R}^n$
Let $X,Y$ be finite-dimensional real normed spaces. Consider the set of linear operators $L(X,Y)$ between the two spaces.
Then we define the set of equivalence classes
$$G(X,Y):=\left\{[T]; T,S \in ...
- 536
4
votes
1
answer
281
views
Weak sequential compactness on the space of compact operators
Let $E,F$ be Banach spaces and let $A\subset K(E,F)$ be a subset of the space of compact operators from $E$ to $F$.
A result by Kalton states that $A$ is weakly compact if and only if $A$ is WOT* ...
- 103
6
votes
1
answer
381
views
Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?
Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$
is not ...
- 395
4
votes
1
answer
364
views
Approximation property of a Banach space in terms of finite-rank projections
Let $X$ be a separable Banach space. Is this property equivalent to the approximation property?
There exists a chain $X_n$ of finite-dimensional subspaces of $X$, each being a range of some ...
- 97
7
votes
2
answers
249
views
Do the operators in $B(E,F)$ separate points on the projective tensor product $F' \mathop{\tilde\otimes_\pi} E$?
Let $E$ and $F$ be Banach spaces, and let $\mathfrak L_{co}(E,F)$ denote the space of bounded linear operators $E \to F$ equipped with the topology of uniform convergence on the absolutely convex ...
2
votes
0
answers
1k
views
Bounded weak and weak-$\star$ topologies and metrics
Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $\overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set
$$
d(x,y) := \sum_{n=1}^\infty 2^{-n} |(x-y, h_n)|...
- 349
4
votes
1
answer
352
views
Minimality properties of James' space
I am interested in the following question about James' quasi-reflexive Banach space $\mathcal{J}$:
Does there exists a non-Hilbertian subspace $X$ of $\mathcal{J}$ such that $X$ isomorphically ...
- 1,005
1
vote
1
answer
194
views
Strictly increasing functions in reflexive subspaces of $C([0,1])$
By the Banach-Mazur theorem, every separable Banach space $X$ embeds into $C([0,1])$. When $X$ is reflexive, it is not possible to find a sequence of disjointly supported, non-negative functions in ...
- 97