# Tagged Questions

A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

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### Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space?

I first specify the setting and then formulate the question precisely. (A very long post follows.) Definitions 1. For $E$ a (real Hausdorff) locally convex space, say that $E$ is suitable iff there ...
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### Closed-form expressions for dual norms of real normed vector spaces

Didn't get any biters over at MSE, so I figure this place might be more appropriate... Say that $V$ is a finite-dimensional real normed vector space, where for some $v \in V$ the norm is notated by ...
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### Maximum Principle with Banach Control Space

This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in ...
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### Orthonormal basis in $\ell^n_p$

Given a $k$-dimensional subspace in $\ell^n_p$, is there a way to bound the value of $$\sum_{i=1}^k \|a_i\|_{\ell^p}^2$$ for $a_i$ an orthonormal (for the "standard" underlying $\ell^n_2$) basis. ...
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### Volume Function on Banach Spaces

I'm looking for a reference for the following so-called Volume Function $V_n$, which is intended to be a Banach/normed vector space generalization of the determinant. Let $X$ be a Banach space with ...
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### Does every $\alpha$-normal ordered Banach space have minimal upper bounds?

Let $\alpha>0$ and $X$ be an $\alpha$-normal (meaning, for $x,y\in X$, $0\leq x\leq y$ implies $\|x\|\leq\alpha\|y\|$) ordered Banach space with closed generating cone $X_{+}$. If $X$ is reflexive, ...
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### Hilbert metric of a sum of cones

Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...
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### extending compact operators to c0

Lindenstrauss has the following paper: http://www.ams.org/journals/bull/1962-68-05/S0002-9904-1962-10787-3/S0002-9904-1962-10787-3.pdf I would like to see the proof for the following theorem (from ...
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### Uniqueness up to isometric isomorphism of predual of $(\sum_{\lambda\in\Lambda} H_\lambda)_{l_\infty}$ where $H_\lambda$ are Hilbert spaces

This fact is an easy consequence of results of the paper by Leon Brown and Takashi Ito, but it looks like an overkill. Does anyone know a simpler proof?
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### Do the banded operators check the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
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### Contractively complemented subspaces of $c_0(I)$

Does every contractively complemented subspace of $c_0(I)$ is isometric to $c_0(J)$ for some $J\subseteq I$? May be someone has a counterexample?
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### Is there Ultracoproduct-like construction for topological spaces in general?

In http://arxiv.org/pdf/math/9704205.pdf they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...
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### What Approximation Property does the space of Schatten-p class operators have?

Background This is a follow-up question to: What (classes of) Banach spaces are known to have Schauder basis? In the previous question, I asked about what spaces are known to have Schauder basis. It ...
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### binary intersection property in finite dimension

I have recently discovered a survey article called "The Hahn-Banach Theorem: The Life and Times" by Narici and Beckenstein and I have read it with great amusement: I was particularly fascinated by ...
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### Weak convergence in the intersection of Lebesgue spaces or Sobolev spaces

Let $B:=B_1\cap B_2\cap...\cap B_n$, where each $B_j$ is a reflexive Lebesgue space or Sobolev space (such as $L^4$, $H^1$, etc.) on a domain in $\mathbb{R}^d$. Then $B$ is a Banach space endowed with ...
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### Agreement of two topologies on a linear space

I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide. Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...
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### What (classes of) Banach spaces are known to have Schauder basis?

Motivation: I am trying to see for what class of Banach spaces the following result is true: There exists an increasing sequence of finite dimensional subspace {$V_n$} of a Banach space X (with ...
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### Direct proof of the separation theorem of Hahn-Banach

The "extension" (or "analytic") form of the theorem of Hahn-Banach has a natural and yet elegant proof. In just any textbook I have ever seen, it is proved first; the "separation" (or "geometric") ...
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### series representation in injective tensor products

All books on tensor products of Banach spaces contain the well-known theorem of Grothendieck that every element of the completed projective tensor product $X \tilde{\otimes}_ \pi Y$ has a ...
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### Continuity of lattice operations in Banach lattices

Let $L$ be a Dedekind-complete Banach lattice. Let $\mathcal{B}$ be the family of nonempty norm-compact subsets of $L$ that are bounded from below. Endow $\mathcal{B}$ with the topology induced by ...
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### $x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology

This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...
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### $(1+\epsilon)$-injective Banach spaces, complex scalars

It is well known that a real Banach space which is $(1+\epsilon)$-injective for every $\epsilon >0$ is already 1-injective (Lindenstrauss Memoirs AMS, 1964, download here). Using common ...
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### almost projective Banach space, complex scalars

It is well-known that if a real Banach space $E$ is "almost metrically projective" then $E$ is isometrically isomorphic to some $\ell^1(\Gamma)$. We say $E$ is "almost metrically projective" if ...
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### Is scalarwise measurability determined by the strong dual?

Since this question has not received an answer so far, I try to reformulate the question in a simpler manner as follows: Do there exist $E,F,\ell,f$ such that $E$ and $F$ are separable (real) ...
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### Counterexample of non-negative sequence weakly converging in $\mathscr{M}^1$ but not $L^1$

Hi. Consider a a sequence of non-negative functions $(f_n)_n$, bounded in $L^1([-1,1])$ and weakly$-\star$ converging in $\mathscr{M}^1([-1,1])$ to some $f\in L^1([-1,1])$. What I mean by this ...
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### Categorical notions involving $\ell_p$ spaces.

First of all, apologies for a somewhat vague question but let me give a try. We know what the projective objects in the category of Banach spaces are: these are precisely $\ell_1(\Gamma)$-spaces. (One ...
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### Special kind of operators

Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation $$A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$ where ...
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### 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 ...
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### Martingale-cotype vs cotype on super-reflexive spaces

I'm have difficultly nailing down the direction of some implications. For $2 \leq q < \infty$, there are (at least) two ways to say that a Banach space $B$ has "cotype $q$". $B$ has cotype q. ...
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### What is the doubling dimension of convex functions?

I am interested in the complexity of convex functions, specifically the "doubling dimension" of the class of convex functions defined on a compact subset of Euclidean space, when compared using the ...
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### Are compact sets in a Banach lattice order bounded?

Given a compact subset $A$ of a Banach lattice $E$, is the following true? There exist $u,v\in E$ so that $u\leq a\leq v$ for all $a\in A$. This is true in case $E=C(X)$, $X$ compact, with the ...
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### Extensions of Carathéodory's theorem

We know about the Carathéodory's theorem which is on the convex bodies of $\mathbb{R}^d$. My question is, how far we can extend it? Is it true for say, any convex object of Banach space, or for convex ...
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### isomophism, commutator

Let X be a Banach space. $B(X)$ is the algebra of all bounded linear operators on X. $\phi: B(X)\rightarrow B(X)$ is a isomoprphisn, $\varphi: B(X)\rightarrow B(X)$ is a isomorphism or negative ...
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### Finite codimensional subspaces of L(X,Y)

Let $X$ and $Y$ be separable Banach spaces and $L(X,Y)$ be the Banach space of bounded linear operators from $X$ to $Y$. Suppose $A$ is a norm closed finite codimensional subspace of $L(X,Y)$. My ...
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### On hyperplanes of $L\infty$

Consider the hyperplane $H=\{f\in L^\infty: \int f = 0\}$ of $L^\infty = L^\infty[0,1]$. My question is: 1. What is the Banach-Mazur distance between $H$ and $L^\infty$? Are there "natural" ...
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### B(H) as a direct sum of a closed two sided ideal and a subalgebra

Let $B(H)$ is the C*-algebra of all bounded linear operators on Hilbert space $H$. Are there a closed two-sided ideal $I$ and a subalgebra $A$ of $B(H)$ such that $B(H)=I \oplus A$ (direct sum I and ...
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### Self-dual finite-dimensional complex normed spaces

Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space? Remarks: There are easy counterexamples in the real case, and in ...
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### Quotients with unconditional bases

Gowers' dichotomy theorem asserts that every Banach space either contains an HI subspace or a subspace having an unconditional basis. There are examples of HI spaces without quotients having ...
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### Is this result of Spain correct?

Let us have a look on the proof of Theorem 2 in [P. G. Spain, Boolean algebras of projections, Proceedings of the Edinburgh Mathematical Society (Series 2) 19, 03, March 1975, 287-289] The author ...
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### Complexifying a real Banach space and its dual

A standard way to define the "complexification" $E_\mathbb{C}$ of a real Banach space $E$ is to define a complex linear structure on $E\times E$ by (1) $(x,y)+(u,v)=(x+u, y+v)$, (2) ...
### Extension of $S_+$ type operators
Let $X$ be a reflexive Banach space and $G\subset X$ a open bounded set. Let $F:\overline{G}\rightarrow X^\star$ be a $S_+$ operator, i.e., if for any sequence $x_n$ in $G$ for which ...
Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...