13
votes
Accepted
interiors of positive cones in ordered Banach spaces
I'll use the notion "ordered Banach space" to denote a Banach space $E$ that is ordered by a closed (and convex) cone $E_+$.
Generally speaken, having non-empty interior is not a common ...
- 12.6k
10
votes
Accepted
Are lattice operations in a Lipschitz space sequentially continuous in the weak* topology?
Yes. If $f_n \to f$ weak* then the sequence $(f_n)$ must be bounded in ${\rm Lip}_0(X)$ (Banach-Steinhaus), and for bounded nets weak* convergence is the same as pointwise convergence. So $f_n \to f$ ...
- 42.8k
7
votes
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
I believe the following is a simple counterexample:
Let $(X,\mu)$ be $\mathbb{N}$ with the counting measure (so I will be writing $\ell^2$ for $L^2(X,\mu)$. Let $g=0$ and $\tilde g(n) = (-1)^n$. Let ...
- 32.5k
7
votes
Accepted
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
The answer is no and the following result provides a quite interesting counterexample. This is a known result, but I am not sure where to find it in the literature.
Theorem. If $f\in L^1_{\rm loc}(\...
- 28k
6
votes
Accepted
Complemented subspaces of Lorentz sequence spaces?
Question 2 has a negative answer. Suppose $\ell_p$ contains uniformly complemented copies $E_N$ of $\text{span}(d_n)_{n=1}^N$. Take an ultra power to get a copy $E$ of the completion of $\text{span}(...
- 31.5k
6
votes
Accepted
Reference request: Spectral properties of real operators
A few preliminary remarks:
1) Complexifications of Banach lattices are in fact a special case of the more general concept of complexifications of real Banach spaces.
2) Most books and articles about ...
- 12.6k
5
votes
Accepted
Subspaces of $C_0$ on which $p$-norm are equivalent?
Use $\|f\|_p \leq \|f\|_\infty^{1- \frac 2p}\|f\|_2^{\frac 2 p}$ to get $\|f\|_\infty \leq C\|f\|_2$ from $\|f\|_\infty \leq C\|f\|_p$.
- 6,035
5
votes
lattice suprema vs pointwise suprema
The answer also depends on the Banach lattice (even if the lattice order is pointwise order). If the functions from $\mathcal{F}$ are pointwise bounded, then the pointwise supremum exists, but the ...
- 1,027
5
votes
Accepted
Reference request: Irreducible operators
Preliminary remarks:
In the comments the OP noted that he is primarly interested in the question whether the dual operator of an irreducible operator is irreducible, so I will focus on this aspect ...
- 12.6k
5
votes
Accepted
Copies of $c_0$ in $C[0,1]$ and disjoint sequences
Take sequences of clopen sets $M_i$, $N_i$, s.t. any two of the $N_i$ have non empty intersection but any three have empty intersection, and the $M_i$ are pairwise disjoint and disjoint from the $N_i$....
- 31.5k
4
votes
Is a Banach lattice isomorphic to a Hilbert space in fact a Hilbert lattice?
Yes. This is a consequence of the fact that semi normalized unconditionally basic sequences in a Hilbert space are Riesz bases. For a theorem quoting proof of what you want, note that that the lattice ...
- 31.5k
4
votes
Accepted
Approximating continuous functions from $K\times L$ into $[0,1]$
The following provides a construction of $g_i, h_i$ satisfying the required conditions (the ranges of these functions can even be in $[0, 1]$):
Claim: For each $k \in K$, there exists an open ...
- 2,830
3
votes
When do positive operators have eigenvalues?
Here is one result that could, sometimes, be helpful in the setting of the question:
Theorem. Let $(\Omega,\mu)$ be a finite measure space and let $0 \not= T: L^2(\Omega,\mu) \to L^2(\Omega,\mu)$ be a ...
- 12.6k
3
votes
Accepted
Weak compactness of order intervals in $L^1$
As Jochen Wengenroth mentioned in the comments, the weak compactness of order intervals follows from their uniform integrability. I think the proof of the Dunford-Pettis theorem is reasonably ...
- 1,027
3
votes
Accepted
Reference on inductive (direct) limit of ordered vector spaces and vector lattices
There is a paper by Wolfgang Filter form 1980s.
W. Filter, Inductive limits of Riesz spaces, Proceedings of the International Conference held in Dubrovnik, June 23-27, 1987 (Bogoljub Stankovic, Endre ...
- 46
3
votes
The complement of $L_1(0,1)$ in $L_1(0,1)^{**}$
If I understand your question correctly, you are interested in a convenient description of $L_1^d(\mu)$. The finite additive measures are very inconvenient to work with. But Gelfand's representation ...
- 61
3
votes
Accepted
Eigenvectors of the dual of positive irreducible operators
No, the dual operator $T'$ does not have a positive eigenvector, in general.
As a counterexample, consider the space $E = c_0(\mathbb{Z})$ if scalar-valued sequences indexed over the integers, endowed ...
- 12.6k
2
votes
Accepted
Is a certain property of a continuous map preserved under "surjectification"?
The answer here is negative: for $Y$ take the remainder $\beta\omega\setminus\omega$ of the Stone-Cech remander of the discrete space $\omega$ of finite ordinals. In the space $Y$ take any countable ...
- 42k
2
votes
Accepted
Looking for a paper on axiomatic orthogonality in a vector space
This journal published by the Herzen University is not yet available in electronic form.
A paper version can be found in multiple libraries, including the National Library of Russia.
They will scan ...
- 37.8k
2
votes
How to characterize the order convergence in Bochner-integrable functions space?
"If $f_n\to 0$ a.e., then $g_n\to 0$ a.e.? Is this true? (*)"
No, this is not true in general. E.g., suppose that (i) $\mu$ is the only probability measure over the singleton set $\Omega:=\{...
- 128k
2
votes
Accepted
Is the union of good equivalence relations on a compact space good?
It seems that the quostion about the skeletal property of $\psi$ has negative answer.
Let us recall that a map $f:X\to Y$ between topological spaces is skeletal if for any nonempty open set $U\...
- 42k
2
votes
Accepted
Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?
It is not true. Let $X = \beta \mathbb{N}$, so that $C(X) \cong l^\infty$. For each $i, k\in \mathbb{N}$ let $f_{i,k}$ be the function which is constantly $1$ on $\{1, \ldots, i\}$ and constantly $\...
- 42.8k
2
votes
Accepted
Decomposition of weak* convergent nets into positive weak* convergent nets
Based on the comment by @BillJohnson:
If $0\le \omega\in E$ then $\|\omega\| = \omega(e)$ implying that a net of positive elements in $E$ converges to $0$ in weak$^*$ topology if and only if it ...
- 71
1
vote
Is the Lorentz space $L_{W,1}(0,1)$ isomorphic to $L_1(0,1)$?
The answer, as expected, is no. There must be several ways to prove it. Corollary 2.e.8 in Lindenstrauss-Tzafriri's book (vol. 2) provides one.
- 61
1
vote
lattice suprema vs pointwise suprema
Theorem. Let $\mathcal{F}$ be a class of measurable functions defined in a
measurable set $E\subset\mathbb{R}^n$. Then $\bigvee\mathcal{F}$
exists and there is a countable subfamily
$\mathcal{G}\...
- 28k
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