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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Something is wrong with this argument (Lorentz and Rosenthal-Woo sequence spaces)

Fix once and for all $1<p<\infty$. Throughout, $w=(w_n)_{n=1}^\infty$ will denote a sequence of positive real weights satisfying \begin{equation}1=w_1\geq w_2\geq w_3\geq\cdots>0\;\;\;\text{ ...
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Norm of a unital homomorphism

Let $\mathcal{A}_1$ and $\mathcal{A}_2$ be two unital $C^*$-algebras and $\varphi : \mathcal{A}_1 \to \mathcal{A}_2$ a unital $*$-homomorphism, i.e. a linear map such that $\varphi(xy)=\varphi(x)\...
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Showing that $\log (\log 1+\frac{1}{|x|})$ belongs to $W^{1,p}(\Omega)$ for $p \geq 2$.

I want to show that the function $f$ belongs to $W^{1,p}(\mathbb{R}^n)$ for $p \geq 2$, where $f$ is defined as $$ f(x)=\log\left(\log \left(1+\frac{1}{|x|}\right) \right)$$ Note: This is an example ...
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32 views

Bounding $L_p$ norms on a convergent $L_1$ sequence

I've encountered a prelim problem on $L_p$ spaces that I'm pretty stuck on. Suppose $1 < p < \infty$ and $f_n \in L_1([0,1]) \cap L_p([0,1])$, with $||f_n||_p$ bounded above by some constant $M$...
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16 views

Relationship between spectral rays commuting

Show sing binomial Newton that if $S$ and $T$ are continuous operators on Banach space $B$ such that $ST = TS$ then $$r_\sigma(T + S) \leq r_\sigma(T) + r_\sigma(S)$$ where $r_\sigma$ is spectral ...
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1answer
26 views

Showing that if $E^*$ is reflexive, then $E$ is reflexive , for a Banach space $E$. [duplicate]

Let $E$ be a banach space. I have already shown that if $E$ is reflexive then $E^{*}$ is reflexive. Now I want to show that if the dual space $E^{*}$ is reflexive, then $E$ is reflexive. If $E^...
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12 views

Putting a Lower Bound On The Power Type Of Modulus Of Uniform Convexity

Take a Banach space $(X,\|\cdot\|_X)$ and define, on the interval $[0,2]$, the modulus of uniform convexity to be $\delta_X(\epsilon) = \inf\{1-\frac{\|x+y\|}{2}:\|x\|=\|y\|=1, \|x-y\|=\epsilon\}$. $...
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89 views

Continuous injection and density in $l_p$ spaces

If $r \le s$ then $l_r$$\subseteq$ $l_s$ . How can I prove there is a continuous injection $l_r$ $\hookrightarrow$ $l_s$? The suggestion was to use the fact that $\Vert$x$\Vert$$_r$ $\le$ $\Vert$x$\...
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41 views

What does well-isomorphic mean?

What I'm currently reading discusses the notion of spaces being well-isomorphic, specifically a Banach space containing well-isomorphic copies of $\ell_1^n$ for every $n\in\mathbb{N}$. The author ...
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Restriction of operators on $l_\infty$ to $c_0$

Given $\epsilon>0$, can we always find a non-compact operator $T:l_\infty\to l_\infty$ of norm larger than $1$ such that the restriction of $T$ to $c_0$ is compact and has norm smaller than $\...
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43 views

$B$ be an uncountable dimensional real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear?

Let $B$ be a real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear ? I know that it is true if $B$ is a Hilbert space ( we only need an inner product ...
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36 views

In $(C[0,1],||.||_{\infty})$ , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded?

Consider the metric space $(C[0,1],||.||_{\infty})$ , in this space , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded ? Please help , Thanks in advance
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subspace of approximation property

A Banach space $X$ has the approximation property (AP) if for any compact subset $K$ of $X$ and any $\epsilon>0$, there exists a bounded finite rank operator $R$ such that $\| x - R(x) \| < \...
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50 views

Proving continuity of a operator $T\colon E \to E'$ [duplicate]

Let be $E$ a Banach space over real numbers and $T\colon E \to E'$ linear such $T(x)(x)\geq 0$ for all $x\in E$, prove T is continuous. If $x_n\to x$ and $T(x_n)\to \phi\in E'$ then $T(x_n)(y)\to\...
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47 views

Norms on an Ultraproduct

Suppose $X$ is a Banach space and $\mathcal{U}$ is a non-principal ultrafilter on $\mathbb{N}$. I am interested in the Banach space $(X)_\mathcal{U}$, where we consider sequences $(x_i)_{i \in \mathbb{...
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2answers
35 views

Space of Lipschitz continuous functions is complete

Let $X$ be set of functions $f:[-1,1]\to \mathbb{C}$ such that $f(0)=0$ and there exists $\alpha>0$ such that $$ |f(t)-f(s)|\le \alpha |t-s| $$ for all $t,s\in [-1,1]$. Equip $X$ with the norm: $...
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22 views

Bounded linear extension of an operator whose co-domain is incomplete.

Let $X$ be a normed space and $Y$ be a Banach space. The bounded linear extension theorem states that a bounded linear operator $T : M \to Y$, where $M \subseteq X$ is dense, can be extended to a ...
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21 views

dom(A) is a Banach space w.r.t. the Graph-norm

Let $X$ and $Y$ be Banach spaces and let $A:dom(A)\to Y$ be a linear operator, defined on a linear subspace $dom(A)\subset X $. Prof that the graph of $A$ is a closed subspace of $X\times Y$ if and ...
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20 views

A singleton as domain sum of a series

Consider the series $$e_1+\frac{1}{2}e_2-\frac{1}{2}e_2+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{4}e_3-\frac{1}{4}e_3+\frac{1}{8}e_4-\frac{1}{8}e_4+\cdots-\frac{1}{8}e_4+\frac{1}{16}e_5-\cdots$$ in the ...
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60 views

Trouble finding a function satisfying an integral equation

I'm stuck at the last step of this exercise: b) Use the Banach fixed point theorem to show that there is a unique function $f \in C[0,1]$ for which the equation $$f(t) + \int_0^1e^{\tau+t-3}f(\tau)...
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Is it true that the closed convex hull of a compact subset of the dual equipped with the w*-topology is compact?

Let $X$ be a Banach space. Consider the dual $X^*$ equipped with the weak*-topology. Is it true that the closed convex hull of a compact subset $K$ of the dual $X^*$ is compact? ps: I know that the ...
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39 views

extending a functional while preserving its norm

Suppose $E$ is a finite-dimensional subspace of a Banach space $X$, and $x_0\in X$ is a vector with $x_0\notin E$. Suppose $f\in X^*$ is a continuous linear functional and that $a>0$ is a positive ...
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47 views

How to find out if a function belongs to $H^2$ or $H^1$

I'm beginning with Sobolev spaces and I found out, that $$ H^k = W^{k,2}. $$ I've also seen the following exercise recently: $$ \frac{1}{2}u'' = 1 $$ And here I'm supposed to find out if $u$ ...
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1answer
53 views

When is $C(X)$ reflexive?

Let $X$ be a compact Hausdorff space. What are sufficient and necessary conditions on $X$ under which $C(X)$ would be a reflexive Banach space. Is there a non reflexive Banach space $C(X)$ such that $...
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21 views

Analyticity of a certain Banach manifold

I am trying to make sense of some infinite dimensional topology. Let $\mathcal{A}_r$ to be the set of all real-analytic functions $\mathbb{C}^2\to\mathbb{C}^2$, whose radius of convergence is at ...
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Bijection from linear subspaces extends to their closures

Let $E,F$ is Banach spaces, and $\pi : E \to F$ is linear bounded surjective operator. Let $N$ is linear subspace (not necessary closed!) of $E$, $M$ is linear subspace (not necessary closed too) of $...
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Showing that $(a_n)_n \in l_1$ provided $\sum_{k=1}^\infty a_kx_k$ exists for any $(x_n)_n \in c_0$

I tried first using the fact that $c_0$ is Banach to apply the Uniform Boundedness Principle on the function series $(T_n)_n = \{\sum_{k=1}^n a_kx_k\}$, $T_n:c_0 \rightarrow \mathbb{K}$, and then to ...
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35 views

spectrum of an operator restricted to an invariant subspace

Let $X$ be an infinite-dimensional real Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. Suppose $W$ is a finite-codimensional $T$-invariant closed subspace of $X$, ...
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1answer
19 views

Every Cauchy sequence in $\{f\in (C([0,1]),\|\cdot\|_1)\,|\,\exists a,b\in\mathbb R:f(x)=ax+b\}$ converges

I have trouble proving that, using the norm $\|f\|=\int_0^1|f(x)|\mathrm dx$, for a Cauchy sequence of functions $f_n(x)=a_nx+b_n$, the sequences $(a_n)_n$ and $(b_n)_n$ also have to be Cauchy ...
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The set of w*-continuous operators is closed for the weak* topology?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...
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42 views

A problem of nets in topology

Let $G$ be a topological group with neutral element $e$. Let $\pi \colon G \to B(E)$ a (non-continuous) representation of $G$ on a Banach space $E$ by bounded linear operators. Let $T$ an element of ...
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1answer
20 views

Vectors in Normed Space Must Have Finite Length?

I have assumed this to be the case, and consequently this is why one looks at convergent sequences of vectors in normed, Banach, and Hilbert spaces. But, I've never seen this listed explicitly as an ...
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77 views

Bijectivity and Lipschitz continuity of a function on a Banach space

I don't really know how to solve the following exercise, I need a little help: a) Let $(X,\|\cdot \|)$ be a Banach space and $F\colon X \to X$ Lipschitz-continuous (i.e. $|F(x) - F(y)| \le L|x-y|$)...
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Applying a functional over an infinite sum

The space $\ell^p$ has a Schauder basis and so we can uniquely express any element of $\ell^p$ as: $$x= \sum^\infty_{i=1}{\alpha_i}{e_i}$$ Where $(e_i)$ is the Schauder basis for $\ell^p$. My ...
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Peano's theorem , initial value problem, Banach Spaces

I'm Taking a Course in Differential Equations and this is one of the exercises those I have to do at home, I can't come up with these short questions: Let X be an infinite-dimensional Banach space. ...
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54 views

Let $X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,\ f(0)=0\right\}$. Show $(X,\lVert\cdot\rVert_X)$ is complete.

The following is a problem on an old Analysis preliminary exam at my institution; I'm prepping for the prelim. The problem is: Let $$X=\left\{ f:[0,1]\to\mathbb R \;\big|\; \lVert f\rVert_X<\infty,...
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20 views

Does the norm of an operator on a $w^{*}$ dense subspace determine its norm?

Let $X$ be a (separable) Banach space, $T:X^{*}\to X^{*}$ a bounded operator, and $Y\subset X^{*}$ a norm closed, $w^{*}$ dense subspace of $X^{*}$. Is it true that $\|T\|=\|T_{|Y}\|$?
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31 views

Is there a Banach space which is not isometrically isomorphic to $l^p$?

I know that every Hilbert space is isometrically isomorphic to $l^2(\beta)$ where $\beta$ is a Hilbert basis for that space. Do Banach spaces have the similar property? That is, is every Banach space ...
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1answer
23 views

If $\varphi \in E'$ and $A$ is convex and open then $\varphi (A)$ is an open interval

Let $E$ be a real normed space and $\varphi \in E'$, $\varphi \neq 0$. Suppose that $A \subset E$ is an open convex not empty subset. Show that $\varphi(A)$ is an open interval. Since $A$ is ...
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28 views

Closable Operators: Nonexample

Given the Banach space $X:=\mathcal{C}([0,1]\cup[2,3])$. I remember I've seen a beautiful example of a non-closable operator whose graph is dense. It involved exploiting Stone-Weierstraß for a ...
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1answer
22 views

Hahn-Banach: Operators

Given two Banach spaces $X$ and $Y$. (More generally locally convex spaces) Regard a closed subspace $U\subseteq X$. Does every bounded operator extend: $$T\in\mathcal{B}(U,Y)\implies T_E\in\mathcal{...
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35 views

On Schauder basic systems in universal enveloped algebra of system of countable family of bounded selfadjoint operators

Let $A = C^*(1,T_1,T_2, ... | T_i^* = T_i, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of selfadjoint operators. I want to know as more as possible about that algebra,...
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28 views

If $A$ is bounded and dissipative, is $\lambda \mathbb 1-A$ invertible for $\lambda>0$?

Let $X$ be a Banach space and $j$ a map (not necessarily linear or anti-linear!) from $X$ to $X'$ so that $$j_x(x)=\|x\|^2 \qquad \forall x \in X \text{ and }\qquad \|j_x\|_{X'}=\|x\|$$ We say that ...
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37 views

Proof that $C(K)$ is a Grothendieck space for $K$ an extremely disconnected compact space.

I am looking for a proof, other than the original article by Grothendieck which is in French, that the space $C(K)$ is Grothendieck when $K$ is extremely disconnected.
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24 views

Formal series which uniformly bounded in each representation of universal $C^*$ algebra converge

Let $A = C^*(T_1,T_2,...|T_i^* = T_i, ||T_i||\leqslant 1)$ - universal $C^*$ algebra of countable family of selfadjoint operators. I have formal series $x = \sum_{i_1,...,i_k} \alpha_{i_1, ..., i_k} ...
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1answer
29 views

Quotient maps in Banach spaces

I came across two definitions of quotient maps in Banach spaces. A bounded linear transformation $T:X\to Y$ is a quotient map if: A) $T(\textrm{int}(B_X))=\textrm{int}(B_Y)$ B) $\overline{T(B_X)}=...
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52 views

Unit ball of $L^1$, $L^\infty$ and $C(X)$ is not strictly convex

I need to show that the unit balls of $L^1(\mu)$, $L^\infty(\mu)$ and $C(X)$ are not strictly convex. I have already shown that if $1<p<\infty$ then the unit ball of $L^p(\mu)$ is strictly ...
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47 views

Weak convergence of Banach space valued random variables

In the book 'Probability in Banach Spaces: Isoperimetry and Processes', available here http://michel.talagrand.net/, in the second chapter on the page 34, above the Theorem 2.1 there is a statement. (...
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25 views

The quotient norm on $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$.

I try to show that the norm on the quotient space $\ell^{\infty}(\mathbb{N}) / c_0 (\mathbb{N})$ is given by $\limsup_{n \in \mathbb{N}} |x_n|$, where $x = (x_n)_{n \in \mathbb{N}} \in \ell^{\infty} (\...
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1answer
41 views

Exercise 1.65 of Megginson's “An Introduction to Banach Space Theory”.

Unfortunately I do not succeed in completing the following exercise: Let $X$ be a Banach space and let $T : X \to \ell^{1} (\mathbb{N})$ be a linear operator. For each $n \in \mathbb{N}$, let $(Tx)...