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

learn more… | top users | synonyms

37
votes
0answers
2k views

Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
7
votes
0answers
145 views

defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
6
votes
0answers
71 views

The dual of the Banach space $C(\Omega)$

It is well-known that the dual of the Banach space $C([0,1])$, i.e. the space of all continuous functions on the interval, is the space of all functions of bounded variation on the interval, ...
6
votes
0answers
141 views

Reference request for the fact

Does anyone know a reference to the paper or a textbook where this fact is proved $$ \mathcal{B}(\bigoplus_1 X_\alpha, Y)\cong_1 \bigoplus_\infty \mathcal{B}(X_\alpha, Y) $$ Most author are bored to ...
6
votes
0answers
146 views

Linear isomorphisms with dense graph

Is it true that for each infinite dimensional Banach space $X$ there exists a linear bijection $f: X \rightarrow X$ with a dense graph? A graph of $f$ it is the set $\Gamma(f):=\{(x, f(x)): x \in X ...
5
votes
0answers
41 views

Smallest constant of Lipschitz retraction from bounded to continuous functions

Let $B$ be the space of all bounded functions $f:[0,1]\to\mathbb R$ equipped with the supremum norm*. It contains $C$, the space of continuous functions on $[0,1]$, as a subspace. An $L$-Lipschitz ...
5
votes
0answers
49 views

Why should the open mapping theorem be expected?

Soft question alert. I want to know why to expect the open mapping theorem to be true. My thoughts: I know that one nice consequence of the OMT could be thought of as the universal property of ...
5
votes
0answers
70 views

Weak*-complemented subspaces of $\ell_\infty$

Consider $\ell_\infty$ as $\ell_1^*$. Let $X$ be an infinite-dimensional complemented subspace of $\ell_\infty$ (in partiuclar, $X$ is isomorphic to $\ell_\infty$). Can we find a further subspace ...
5
votes
0answers
53 views

a question about Tsirelson's space

Background. Let $T$ denote the Figiel-Johnson construction of the Tsirelson space, that is, the completion of $c_{00}$ under the implicitly-defined norm ...
5
votes
0answers
155 views

Subsequences of a basic sequence

Suppose ($x_n$) is a basic sequence in a Banach space $X$, and $Y$ is a closed, infinite co-dimensional subspace of the closed span of $(x_n)$. Can we always find a subsequence ($y_n$) of ($x_n$) such ...
5
votes
0answers
129 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
5
votes
0answers
162 views

Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...
4
votes
0answers
42 views

Alternate proof of a result on dual spaces: what is wrong with it?

I am familiar with Rudin's book's proof of the fact that, in $\sigma$-finite measure spaces and for $p\in[1,+\infty)$, the dual space of $L^p$ is $L^q$ where $p,q$ are conjugate, i.e. ...
4
votes
0answers
48 views

Conditions for Taylor formula

I know that, if $F:X\to Y$, where $X,Y$ are Banach spaces, is a map whose $n$-th Fréchet derivative $x\mapsto F^{(n)}(x)$ is continuous as a function of $x$ in a neighbourhood of $x_0\in X$, then the ...
4
votes
0answers
62 views

finding the algebraic dimension of $\ell^p$ spaces

I want to know "how we can find the algebraic dimension(the cardinal number of the Hamel basis) for $\ell^p$ spaces." What can we say about $\ell^p(I)$, where $I$ is an infinite set?\ Moreover, for ...
4
votes
0answers
48 views

Conditional expectation onto maximal abelian subalgebras

If you take a von Neumann algebra $M$ and any its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take ...
4
votes
0answers
108 views

How to get a grip on codimensions

I am trying to find a proof for the following problem: Let $X,Y$ be Banach spaces $A,B:X \rightarrow Y$ are bounded linear operators $Ran(A)$ is closed, and $\dim(\mathrm{Ker}(A))$ or ...
4
votes
0answers
64 views

Equivalence of definitions of $C^k(\overline U)$

let $U$ be an open set of $\mathbb{R}^n$, that contains at least some open set. In Evans book we find the definition $$C^k(\overline U)=\{f \in C^k(U): D^\alpha f \text{ is uniformly continuous on ...
4
votes
0answers
88 views

Ultraweak topology on Banach spaces

If $X$ and $Y$ are Banach spaces with $Y$ reflexive, then the space $\mathcal{B}(X,Y)$ of bounded operators from $X$ to $Y$ is the dual of the projective tensor product of $X$ and $Y^{*}$. As in the ...
4
votes
0answers
124 views

Non linear compact map

Suppose to have two Banach spaces $E$ and $F$, with $E$ reflexive. Suppose to have a continuous map $T:E \to F$ which maps bounded subsets into precompact subsets. $T$ is not assumed to be linear. ...
4
votes
0answers
154 views

Non strictly singular operators

Let $X$ be a separable Banach space and let $T:X\to X$ be a bounded operator that is not strictly singular. Can we always find an infinite dimensional, closed, and complemented subspace $Y$ of $X$ ...
4
votes
0answers
105 views

Differential calculus on Banach space

I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly. Problem Given the ...
4
votes
0answers
57 views

A space $X$ that contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and whose dual is not weakly sequentially complete

I want to find an example of a Banach space $X$ which contains a copy of $\ell_1$, does not contain a complemented copy of $\ell_1$, and so that $X^*$ is not weakly sequentially complete.
4
votes
0answers
69 views

Banach dual space integral

Let $X$ be a Banach space and $f_t \in X^*$ for each $t \in [0,t_0]$. Suppose that $$\int_0^{t_0} f_t(x) = 0$$ for all $x \in X$. 1) Does it make sense to write $\int_0^{t_0}f_t = 0$? 2) If so, does ...
4
votes
0answers
125 views

Homeomorphisms on X and automorphisms on C(X)

Let $ X $ be a compact Hausdorff space. Let $ \psi $ be a homeomorphism on $ X $. Let $ \text{Aut}(C(X)) $ be the group of automorphisms of $ C(X) $, and $ \text{Homeo}(X) $ be the group of ...
4
votes
0answers
80 views

The control of norm in quotient algebra

Let $B_1,B_2$ be two Banach spaces and $L(B_i,B_j),K(B_i,B_j)(i,j=1,2)$ spaces of bounded and compact linear operator between them respectively. If $T \in L(B_1,B_1)$, we have a $S \in K(B_1,B_2)$ and ...
4
votes
0answers
109 views

A question regarding vector spaces with partial order

$\newcommand{\N}{\mathbf{N}}$$\newcommand{\R}{\mathbf{R}}$ Let $X=(X,\leq)$ be a Riesz space (a lattice which is also an ordered vector space over $\R$). Define $X^+=\{x\in X\colon x\ge 0\}$. Are the ...
4
votes
0answers
65 views

complemented subspaces of $L_{p}$ spaces (Question posed incorrectly earlier)

This question was asked incorrectly originally in such a way that it probably made no sense. Fixed version: I know that $L_{p}[0,1]$ has $\ell_{2}$ as a complemented subspace, and I'm wondering if ...
4
votes
0answers
89 views

Preduals of Banach spaces and in particular of $\text{BMO}(\mathbf R^d)$

In general the predual of a Banach space is not unique. If there are multiple ones must they be isomorphic? More specifically is $H^1(\mathbf R^d)$ the only predual of $\text{BMO}(\mathbf R^d)$ or ...
3
votes
0answers
19 views

Finding a norm making a subspace dense

Suppose $V$ is a (real or complex) vector space and $W$ is a subspace of $V$. Under what conditions is there a norm on $V$ making $W$ a dense subspace of $V$? That $V$ and $W$ have the same ...
3
votes
0answers
29 views

give necessary and sufficient conditions that every functional in $w^*-cl M$ be the $w^*$- limit of a sequence from M

Let $X$ be a separable Banach space. If $M$ is a linear manifold in $X^*$ give necessary and sufficient conditions that every functional in $w^*-\mathrm{cl} M$ be the $w^*$- limit of a sequence from ...
3
votes
0answers
65 views

Using Stone–Weierstrass theorem for completely regular space

Let $X$ be completely regular. If $K$ is a compact subset of $X$, define $$p_K(f)=\sup\{|f(x)|;x\in X\}$$ then $\{p_k; \text{K is a compact }\}$ is a family of seminorms that makes $C(X)$ into a ...
3
votes
0answers
30 views

Fredholm Integral in Bayesian Appliation

Let $X = x_1, x_2, \ldots, x_n$ be a sequence of Bernoulli random variables with $k$ successes. Suppose that, given $X$, the posterior predictive probability of $x_{n+1} = x$ is known to be $g(x)$ ...
3
votes
0answers
113 views

Asymptotically isometric copy of $\ell_1$

A Banach space $X$ is said to contain Asymptotically isometric copy of $\ell_1$ if there is a null sequence$(\rho_n)_{n=1}^{\infty}$ in $(0,1)$ and a sequence$(x_n)_{n=1}^{\infty}$ in $X$ so that ...
3
votes
0answers
54 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
3
votes
0answers
114 views

Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
3
votes
0answers
173 views

On the weak convergence in reflexive Banach space

Consider the following proposition: Proposition 1. Let $X$ be a reflexive Banach space and suppose that $\{x_n\}$ is a sequence in $X$ that is bounded and has at most one weakly sequentially cluster ...
3
votes
0answers
58 views

The dual of the Annihilator

Let $X$ be a Banach space, and $I$ be a closed subspace. Then it's known that $(X/I)^*=I^{\perp}$. My question is what is the second dual of $X/I$? or what is the dual of $I^{\perp}$ ? If we know ...
3
votes
0answers
98 views

Every closed separable subspace is complemented

Let $X$ be a Banach space. Suppose that every closed separable subspace $Y$ of $X$ is complemented in $X$ (i.e., there is a bounded linear projection of $X$ onto $Y$). Is $X$ necessarilly isomorphic ...
3
votes
0answers
54 views

operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
3
votes
0answers
70 views

why is test function space $\mathcal{A}$ complete

I am trying to find out, why the space $$\mathcal{A}:=\left\{\phi\in C_0(\mathbb{R}^{2d})|\;\|\phi\|_\mathcal{A}:=\int_{\mathbb{R}^d}\sup_{x\in\mathbb{R}^d}|(\mathcal{F}_p\phi)(x,y)|\;\mathrm ...
3
votes
0answers
101 views

Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
3
votes
0answers
80 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
3
votes
0answers
88 views

$M+N$ is a closed subspaces of banach space iff $M^{\bot} +N^{\bot}$ is closed subspace of dual

Let $X$ be a Banach space and let $M,N$ be closed subspaces of $X$. I want to prove that $M+N$ is a closed subspace iff $M^{\bot}+N^{\bot}$ is a closed subspace of $X^{\ast}$ (i.e, dual of $X$). Any ...
3
votes
0answers
97 views

Integration for functions with values in a separable Banach space

Let $(X,\mathcal{M},\mu)$ be a measure space, $Y$ a separable Banach space, and $L_{Y}$ the space of all $(\mathcal{M},\mathcal{B}_{Y})$-measurable maps from $X$ to $Y$ (where $\mathcal{B}$ denotes ...
3
votes
0answers
84 views

Application of a result on some bounded functionals on a subspace of $C([0,1])$

The following result was proved in a previous post: Bounded functionals on Banach spaces. Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], ...
3
votes
0answers
54 views

An example of a space $X$ such that every L-subset of $X^*$ is weakly precompact but not relatively weakly compact

A bounded subset $A$ of $X^*$ is called an L-set if each weakly null sequence $(x_n)$ in $X$ tends to zero uniformly on $A$. The space $X$ has the Reciprocal Dunford-Pettis property if every L-subset ...
3
votes
0answers
105 views

Density of operators

I am interested in operators on non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{**})$ by ...
3
votes
0answers
120 views

Question about proof of completeness of $L^p$

In my notes we prove completeness of $L^p$ by showing that if $\sum\|f_k\|_p < \infty$ then $\sum f_k$ converges in $L^p$. (That's a lemma we prove a bit earlier, namely that $(V, \|\cdot\|)$ is ...
3
votes
0answers
142 views

What is the Dunford Integral and why is it useful?

Wikipedia defines the Pettis Integral for Banach space valued functions on a measure space by duality. Apparently there is a Dunford integral which specializes to the Pettis integral. What is its ...