# Tagged Questions

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### when one can expect $\left \| |x|^{2}x- |y|^{2}y \right \| \leq \frac{1}{2} \left \| (x-y) \right \|$?

Suppose $(X, \left\|\cdot \right \|)$ is Banach algebra with the property that $\left\| (|x|^{2}x) \right\| \leq \left\|x\right \|^{3};$ for every $x\in X.$ Let $y_{0}\neq 0 \in X$ and fix it; and ...
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### Understanding the Frechet derivative

What is the relationship between the normal 'high school' concept of a derivative and a Frechet derivative? According to wikipedia the Frechet "extends the idea of the derivative from real-valued ...
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### The Banach space $c_0$ is $C^{\infty}$-smooth.

In this paper, J. Eells defines this notion of $C^r$-smoothness for Banach spaces: A Banach space $E$ is $C^r$-smooth, $r \geq 0$, if there exists a nontrivial (that is, nonzero) $C^r$ function ...
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### annihilator of an intersection in infinite dimension

Given two subspaces of an infinite dimensional Banach space, is the sum of their annihilators dense in the annihilator of their intersection?
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### How to prove that sequence spaces $l^{p}, l^{\infty}$ and function space $C[a. b]$ are of infinite dimension

I am studying about the sequence space $l^{p}, l^{\infty}$ and function space $C[a. b]$. It is mentioned in the book that all of these spaces are of infinite dimension. I want to prove that these ...
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Let $$c=\{ (a_i)_{i \in \mathbb{N}} \ ; \ \ a_i \in \mathbb{R}\ ,\ \forall i \in \mathbb{N} \ , \ \mbox{exist} \ \displaystyle \lim_{i \to \infty}(a_i)\}$$ $$l^{\infty}=\{ (a_i)_{i \in \mathbb{N}} \ ... 0answers 78 views ### Differential calculus on Banach space I'm revising for my upcoming test, and this problem dated back some years ago. I've been working on this problem for almost a day, but I don't even know how to start it correctly. Problem Given the ... 1answer 31 views ### Is there any space with normal structure but not uniform normal structure? It is clear that uniformly normal structure implies normal structure. But, is there any space that has normal structure but it doesnt have uniformly normal structure?? Would you mind to give me some ... 0answers 25 views ### Lemma 3.2 from “Positive solutions for third order semipositone boundary value problems” How de prove this lemma please : Assume that: w(t) is nondercreasing and w(t)>0 on (q,1] , \frac12<p<q<1 hods . Let z\in C^2[0,1]\cap C^3(0,1) satisfy z'''(t)\geq 0 0n ... 1answer 82 views ### Sum of Banach spaces Let H^2(\mathbb{R}^3) the usual Sobolev space and consider the following set$$X=\bigg\{u\bigg|u=\phi+\frac{Q}{|x|},\phi\in H^2,\,\, Q\in\mathbb{C}\bigg\}$$I observe that the decomposition is ... 1answer 403 views ### Unit ball of a Separable Banach Spaces is metrizable Please help me to understand why the unit ball of a separable banach space is metrizable, when it is given the induced topology from the weak topology. Specifically, my problem is I think I'd be able ... 1answer 113 views ### Strong and weak-* convergence for bounded linear maps The textbook I am using says that a sequence (T_n) in \mathcal{B}(X,Y) for X, Y normed linear spaces converges strongly to T if \lim_{n\rightarrow\infty}T_nx=Tx for every x\in X. The ... 1answer 120 views ### Banach Space continuous function On the Banach space (C([-1,1]), ||\cdot||_\infty )  consider the operator given by (Tf)(x)= \dfrac{1}{3} \displaystyle\int^1_0txf(t)\ dt + e^x - \dfrac{\pi}{3}  1) prove that the mapping is a ... 1answer 55 views ### Evaluating difficult spectrum Can anyone see how to show the spectrum of the bounded linear operator T on l^1 defined by$$T((\alpha_j)) = (\alpha_j - 2\alpha_{j+1} + \alpha_{j+2})$$is the cardioid$$\{(r, θ) : 0 ≤ θ < 2π, ...
Note ($\Omega,A,\mu)$ is a finite additive space. Let $f\in \bar S(A, \mathbb{R})$ and $y\in Y$ where $Y$ is Banach space. Prove $\int _\Omega yf d\mu = y \int _\Omega f d\mu$. Also, for any $X\in A$, ...