Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [banach-spaces]

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

18
votes
0answers
480 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 ...
8
votes
0answers
132 views

Is there a notion of a continuous basis of a Banach space?

If $X$ is a Banach space, then a Hamel basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as a linear combination of elements of $B$. And a Schauder basis of $...
8
votes
0answers
77 views

Space of linear, continuous, hyperbolic functions is open, dense in the set of invertible functions

Let $(X,||\cdot||)$ be a Banach space on $\mathbb{C}$ and $\mathcal{L}(X)$ the set of linear, continuous functions from $X$ to itself. For $T\in\mathcal{L}(X)$, define the norm $||T||_{\mathcal{L}(X)}:...
8
votes
0answers
166 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
8
votes
0answers
163 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 ...
7
votes
0answers
327 views

Density of Banach spaces

I am trying to understand the notion of density in the context of Banach spaces. The density of a topological space is the least cardinality of a dense subset. Thus, a separable Banach space has ...
7
votes
0answers
372 views

Does the Closed Graph Theorem follow from Banach-Steinhaus?

Q: Is there a simple (but perhaps tricky or clever) proof of the Closed Graph Theorem (or the Open Mapping Theorem, or the result I call the Automatic Inverses Theorem below) from the Banach-Steinhaus ...
7
votes
0answers
218 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 ...
7
votes
0answers
133 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 ...
7
votes
0answers
353 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 ...
6
votes
0answers
55 views

Showing that the equation $x_i - \sum_{j=1}^\infty a_{ij}x_j = b_i$ has a unique solution.

Exercise : Consider the infinite-dimensional system of equations : $$x_i - \sum_{j=1}^\infty a_{ij}x_j = b_i, \quad i=1,2,3,\dots$$ We suppose that $b=(b_1,b_2,\dots) \in \ell^\infty$ and that ...
6
votes
0answers
293 views

Why do we want or need cross-norms on tensor products?

If $(E, \|\cdot\|_1),(F, \|\cdot\|_2)$ are Banach spaces, their (algebraic) tensor product $E \otimes F$ is a vector space looking forward to be normed. A norm on a tensor product of vector spaces is ...
6
votes
0answers
451 views

Areas of research / interest in Banach space theory today.

Whilst taking a class in functional analysis I couldn't help but feel Banach space theory was only ever taught as a natural stepping stone towards Hilbert space theory. As to the prominence and status ...
6
votes
0answers
95 views

Is every $C_0(K,X)$ space isomorphic to $C(L,X)$, for some compact $L$?

Let $K$ be a locally compact Hausdorff space and $X$ be a Banach space. Denote by $C_0(K,X)$ the Banach space of all continuous $X$-valued functions defined on $K$ that vanish at infinity, equipped ...
6
votes
0answers
131 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 ...
6
votes
0answers
238 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, $BV([0,1])...
6
votes
0answers
97 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 ...
6
votes
0answers
105 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 $Y\...
6
votes
0answers
227 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 ...
6
votes
0answers
163 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
171 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
138 views

Characterization of Reflexive Banach Space.

Prove that a real Banach Space $X$ is reflexive if and only if each pair of disjoint closed, convex subsets of $X$, one of which is bounded, can be strictly separated by a hyperplane. The theorem is ...
5
votes
0answers
262 views

$L^p$-space is a Hilbert space if and only if $p=2$

Inspired by $\ell_p$ is Hilbert if and only if $p=2$, I try to prove that a $L^p$-space (provided with the standard norm) is a Hilbert space if and only if $p=2$. I already know that every $L^p$-space ...
5
votes
0answers
38 views

blocks of a normalized basis dominated by lp

I would like to know whether the following conjecture is true, possibly with additional assumptions such as unconditionality. Conjecture 1. Suppose $(x_i)_{i=1}^\infty$ is a normalized basis for a ...
5
votes
0answers
88 views

An exercise in Banach Space Theory

I am currently reading a book and while i was reading I came across an exercise : Prove that a Banach Space $X$ has finite dimension if and only if every linear subspace of $X$ is closed. My ...
5
votes
0answers
66 views

Show that no two eigenvectors of adjoint of right shift operator are orthogonal

Let $$T:\ell^2 \to \ell^2$$ is unilateral shift operator, defined by $$T(x_1,x_2,x_3......)=(0,x_1,x_2,x_3.....),$$ then show that $T$ has no eigenvalue. But every $\lambda \in \mathbb{C}$ such that $|...
5
votes
0answers
162 views

Hilbert valued martingales - help with reference

I'm currently studying the theory of SPDEs on the book "Stochastic equations in infinite dimensions" by da Prato, Zabczyk. In the book, the theory of stochastic processes with values on a Banach space ...
5
votes
0answers
183 views

The Alaoglu's Theorem

I am trying to prove the Alaoglu's Theorem. But my professor told me there is something wrong with my proof. Can anyone help me? Thank you! The Alaoglu's Theorem: Let $X$ be a normed space. Then ball ...
5
votes
0answers
70 views

Complemented $\ell_q$ subspaces of $(\oplus_n\ell_p^n)_\infty$

Fix $1\leq p<\infty$, and denote \begin{equation*}(\oplus_n\ell_p^n)_\infty=\left\{\left((a_i^{(n)})_{i=1}^n\right)_{n=1}^\infty:\left(\|(a_i^{(n)})_{i=1}^n\|_p\right)_{n=1}^\infty\in\ell_\infty\...
5
votes
0answers
228 views

Bishop-Phelps theorem

Bishop-Phelps Theorem: If $E$ is a Banach space and $B\subseteq E$ is bounded, closed and convex, then the linear functionals on $E$, which attain their supremum on $B$, are norm-dense in $E^*$. ...
5
votes
0answers
134 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal X,...
5
votes
0answers
117 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 ...
5
votes
0answers
187 views

Properties shared by equivalent norms.

I am interested in knowing about "geometric" properties shared by equivalent norms on a Banach space. Here I mean "geometric" as opposed to topological, and probably in particular with reference to ...
5
votes
0answers
136 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 \begin{equation*}\|x\|_T=\max\left\{\|x\|_{\...
5
votes
0answers
5k views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
5
votes
0answers
180 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 ...
5
votes
0answers
171 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 ...
5
votes
0answers
155 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 ...
5
votes
0answers
144 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], f\...
5
votes
0answers
246 views

Must-read papers in Operator Theory

I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
4
votes
0answers
64 views

From $\sup_{n\in \mathbb{N}} \left|\int_0^\Lambda e^{nx} f(x) dx \right| < \infty$ to $f\equiv 0$

Given $f\in C[0,\Lambda]$ satisfying $$\sup_{n\in \mathbb{N}} \left|\int_0^\Lambda e^{nx} f(x) dx \right| < \infty$$ Prove that $f\equiv 0$ $\,\forall x\in[0,\Lambda]$ I found a weaker ...
4
votes
0answers
128 views

If $f$ is integrable, then $\| f\|$ is also integrable.

As usual, a partition of a compact interval $[a, b]$ is, by definition, an strictly increasing family $\Pi = (t_k)_{k = 0}^m$ ($m \geq 0$) of points in the interval such that $t_0 = a$ and $t_m = b;$ $...
4
votes
0answers
56 views

If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$

Let $$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$ that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$ Question: Let $\|\...
4
votes
0answers
71 views

Are $\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$ equivalents norms?

Do we have that $$\| \Delta u \|_{L^p(\Bbb R^d)} + \| u \|_{L^p(\Bbb R^d)}~~~~and~~\| u \|_{W^{2,p}(\Bbb R^d)}$$ are equivalent norms This results is pretty easy and straightforward for $p=2$ using ...
4
votes
0answers
134 views

Comparaison of two version of Fractional sobolev spaces: what do we have $W^{s,p}(\mathbb{R}^{n})=H^{s,p}(\mathbb{R}^{n})$?

There are two version of Fractional sobolev spaces . Definition1: (Via Galiardo semi-norm) Let $1\leq p\leq +\infty$, $0<s<1$ and $\Omega\subseteq \mathbb{R}^n$ an open set. The fractional ...
4
votes
0answers
154 views

How to define orientation on infinite dimensional vector space

Let $\mathbb{V}$ be a real Banach space (if someone knows the answer for more arbitrary T.V.S. then great). Is there some concept of orientation that one could define on $\mathbb{V}$ that matches the ...
4
votes
0answers
218 views

A weakly closed set in $X$ that remains weakly-star closed in $X^{**}$

I have $X$ a (non-reflexive) Banach space and $B\subset X$ a weakly closed convex subset. I wonder under what additional conditions (other than weak compactness) $B$ remains weakly-star closed in $X^{...
4
votes
0answers
69 views

Why the denseness of norm attaining operator is important?

I plan for studying norm attaining operators, Lindenstrauss property A and B, property $\alpha$, $\beta$, and so on... Before do it, I wanna understand why these properties are important. I know it ...
4
votes
0answers
56 views

Prove $T:X\to X$ is bounded

Let $X$ be a Banach space and $T:X\to X$ be linear operator. Let $A$ be s subset of $X^*$ which separates the points in $X$. Suppose $f\circ T$ is bounded $\forall f\in A$, show that $T:X\to X$ is ...
3
votes
0answers
45 views

Showing that $\inf \{\|u-g(u)\| : u \in C \} = 0.$

Exercise : Let $X$ be a Banach space and $C \subseteq X$ be closed, convex and bounded. Moreover, let $g:C \to C$ be a non-expansive operator, meaning that : $$\|g(u) - g(v) \| \leq \|u-v\| \; \...