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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Need explanation of problem in Temam (convergence, weak derivatives)

Let $V \subset H \subset V$ be Hilbert triple. We have $u_m$ is infinite differentiable from $[0,T]$ to $V$. Suppose $u_m \to u$ in $L^2(0,T;V)$ and $u_m' \to u'$ in $L^2(0,T;V^*)$ Suppose that it ...
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On the existence of a non-negative function on a Banach space whose limit at every point is infinity.

Does there exist a Banach space $ X $ (possibly non-separable) and a mapping $ F: X \to X $ such that $$ \forall a \in X: \quad \lim_{\substack{x \in X \setminus \{ a \} \\ x \to a}} \| F(x) \|_{X} = ...
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1answer
48 views

Is this Nonlinear Autonomous Banach space valued ODE a flow?

I have the following analogue of Picard's theorem for Banach space valued ode's: Let $O$ be an open subset of a Banach space $B$ and let $F$ be a nonlinear operator satisfying the following criteria ...
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1answer
244 views

A problem on bounded invertible linear operator in Banach space

Let $X$ be a Banach space. Let $T : X \to X$ be a invertible linear operator and $M > 0$ be such that $\|T^{-k}\| \le M$ for all $k \ge 1$. Prove that $\inf_ {n\ge1} \|T^n(x)\| > 0$ for all $x ...
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1answer
96 views

How to determine the spectrum on Banach Space [duplicate]

On Banach Space $C[0,1]$, T is a bounded linear operator and is defined by $Tf(x)=\int_0^xf(y)dy$, then how can I determine the spectrum of T? I was hinted to first show $T^nf(X)=\frac1{(n-1)!} ...
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58 views

canonical form of dyadic martingales

Let $(X_k)_{1\leq k \leq n}$ be a Walsh-Paley $L^p$-martingale (a dyadic martingale) with values in a Banach space $X$. Why does there exist a dyadic martingale $(Y_k)_{1\leq k \leq n}$ with the ...
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1answer
70 views

Do I have a Banach space given this wacky norm?

I have a normed space (I'll denote it $C^2[a;b]$) which consists of continuous real functions whose first and second derivatives are also continuous in interval $[a;b]$. $\forall x,y \in C^2[a;b]$ ...
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1answer
310 views

Infinite dimensional constant rank theorem

Suppose you have an analytic map $\phi : E \rightarrow \mathbb{C}^n$, where $E$ is a complex Banach space, and such that the rank of $D \phi$ is constant. Is it true then that the set ...
2
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1answer
52 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 ...
3
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1answer
559 views

Convex Hull of Precompact Subset is Precompact

I'm trying to prove that, if $K$ is a precompact (I've also heard the phrase totally bounded used for this) subset of a Banach Space $X$, then its convex hull is also precompact. I've come across a ...
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Continuity in a Banach Space

This is a past exam question: On the Banach space $\bigg(C\bigg( \bigg[-\dfrac{1}{2}, \dfrac{1}{2} \bigg], ||\cdot||_\infty\bigg) $, consider the operator given by $$T(f)(x)= x+ ...
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Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
4
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1answer
342 views

Linear isometry between $c_0$ and $c$

The following question is an exercise and so I'm just looking for advices and not for answers if it's possible. I have the following sets in $l^\infty$ $$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
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69 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.
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67 views

Definition of a countable direct sum of subspaces of a Banach space

Let $X$ be a separable Banach space and $K\subseteq X$ a subspace. Let $\{H_i\}_{i\in I}$ be a countable collection of subspaces of $X$. Is it correct that $K=\bigoplus H_i$ iff every element $k\in K$ ...
2
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1answer
64 views

Spaces of class $J_\alpha$

This question is about the spaces of class $J_\alpha$. Given three Banach spaces $Z\subset Y\subset X$ (with continuous embeddings), and given $\alpha\in (0,1)$, we say that $Y$ is of class $J_\alpha$ ...
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$M+N$ 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 ...
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1k views

Examples of Banach spaces

Which of the following are Banach spaces? A. The set of all real-valued functions $f$, $g$ which are functions of an independent real variable $t$ and are defined and continuous on the closed ...
3
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1answer
234 views

Can a space have both a conditional and an unconditional basis?

Does there exist a Banach space $X$ which admits both a conditional and an unconditional Schauder Basis? If so, can one find an example in the collection of $\ell^p$ spaces? My thoughts so far: ...
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1answer
256 views

Normed vector spaces and Banach spaces

Let $X$ be a Banach space with norm $||.||$ and let $S$ be a non-empty subset in $X$. Let $F_b(S,X)$ be the vector space of $F(S,X)$ of all functions $f:S \rightarrow X$ such that $\{||f(s)||:s \in ...
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2answers
376 views

Banach spaces and quotient space

Let $X$ be a normed vector space, $M$ a closed subspace of $X$ such that $M$ and $X/M$ are Banach spaces. Any hint to prove that $X$ must be a Banach space?
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Show that the image of $T:l^{\infty}\to l^{\infty}$, $(x_n)_n \mapsto \Big(\frac{x_n}{n}\Big)_n$ is not closed in $l^{\infty}$.

Denote the set of all bounded sequences in $\mathbb{R}$ by $l^{\infty}$, endowed with the sup norm $\lVert \rVert _{\infty}$. Define a map $T:l^{\infty} \to l^{\infty}$ as follows: $$(x_n)_n \mapsto ...
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156 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 ...
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1answer
68 views

How to show $C(0,T;H)$ is Banach space

$H$ is a Hilbert space. How do I show that the Bochner space $C(0,T;H)$ of continuous $H$-valued functions is a Banach space with the following norm? $$\lVert u \rVert = \sup_{t \in [0,T]}\lVert u(t) ...
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134 views

Counterexample for “the sum of closed operators is closable”

I'm looking for a counterexample in a Banach space. I've seen the counterexample at Sum of Closed Operators Closable?, but I don't understand why $A$ and $B$ are closed. Could someone expand on this ...
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2answers
86 views

Need to confirm: Sup Metric $C[0,1]$, question about boundary

For the sup metric, $C[0,1]$. Let $S \subset C[0,1]$ be given by: $$S=\left\{f:[0,1]\to \mathbb{R} \ : \ 0 \leq f\left(\frac{1}{2}\right)<1\right\}$$ The question is simple: is this set open or ...
3
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1answer
118 views

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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133 views

Is $\| x \| = \sup\{ | \sum_{k=1}^n x_k |\colon n\in \mathbb{N}\} $ a norm in $\ell_1$?

For each $ x= (x_n)_{n=1}^\infty \in \ell_1$ set $$\| x \| = \sup\{ \big| \sum_{k=1}^n x_k \big|\colon n\in \mathbb{N}\} $$ Does this define a norm in $\ell_1$?
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211 views

Is $(l^1 ,\|.\|)$ a Banach space?

Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
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1answer
62 views

Does a cofinite dimensional subspace of a subspace remain cofinite dimensional upon taking closures?

Let H be a separable, infinite dimensional Hilbert space. Let X and Y be (not necessarily closed) subspaces such that X is a cofinite dimensional subspace of Y. Let X′ be the closure of X and Y′ the ...
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1answer
159 views

Is this set dense in $H^1(\Omega)?$

Is $$V_1 = \{v \in H^1(\Omega) \;:\;f(v) = 0 \text{ on } \partial \Omega\}$$ dense in $H^1(\Omega)$ with the same norm as $H^1(\Omega)?$ Here $f$ is some linear functional so that $V_1$ is also ...
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1answer
74 views

A basic result about operators on Hilbert space.

I am studying following result. Let $H$ and $K$ be Hilbert spaces and an operator $A \in B(H, K)$, which has closed range. The spaces $H$ and $K$ have the following orthogonal decompositions: $H = ...
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1answer
205 views

Proving $\ell_\infty$ is complete

I start by taking a Cauchy sequence $(a_i)$ in $\ell_\infty$. I denote the terms of $(a_i)$ as $f_1, f_2, f_3, \dots$ and so on. For each $x \in \mathbb{N}$, the sequence $(f_1(x), f_2(x), f_3(x), ...
2
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0answers
144 views

Space of functions that are everywhere differentiable

Define the space $\beta^1([a,b])$ as the space of functions $f : [a,b] \mapsto \Bbb R$ which are everywhere differentiable and whose derivative $f'$ is a bounded function. One equips this space with ...
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1answer
224 views

“Taylor expansion” of a function taking values in a Banach space

For a Banach space $B$, given a function $f: \mathbb R \to B$, we can define its derivative at $x \in \mathbb R$ as $f'$ such that $$ \lim_{h\to 0} \frac{\|f(x+h) - f(x) - hf'(x)\|}{h} = 0 $$ if the ...
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1answer
173 views

$L_p$ spaces and convergence

The Riesz-Fischer Theorem implies that Lp-convergence implies pointwise a.e. convergence of a subsequence. There is an example that shows that the converse may not be true... Let E = [0, 1], $1 ...
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1answer
286 views

Is the unit ball of a separable Banach space itself separable?

If $X$ is a separable Banach space, then do we know that its unit ball has a countably dense subset contained in the unit ball? This isn't obvious to me.
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1answer
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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 ...
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Maximal Ideals of the Wiener Algebra

I'm wondering why the maximal ideals of the Wiener algebra $\mathcal{W}$ are of the form $\{M_z:z\in \mathbb{T}\}$ where $M_z=\{f\in \mathcal{W}\; |\; f(z)=0\}$. Given that the Wiener algebra is a ...
2
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1answer
172 views

(p-q)-Lipschitz continuity of linear function

I have the following linear function $f(x,y,z) = ax + by + cz.$ I need to prove that f() is (p-q) Lipschitz continuous where $p=1$ and $q=\infty$. For a given two points $(x_1, y_1, z_1)$ and $(x_0, ...
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Weak star limit

Let $\Omega = \mathopen]0,1\mathclose[$ and let a function $A_n: \Omega \to \mathbb R$ defined as: $$A_n(x) = \begin{cases}\alpha &\text{if } k \epsilon \leq x < (k+\tfrac{1}{2}) \epsilon \\ ...
3
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2answers
300 views

$l_1$ equipped with the sup norm is NOT a Banach Space

Prove that $l_1 = \{ x = (x_k)_{k\in\mathbb{N}}\subset \mathbb{R};\ \sum_{k\in\mathbb{N}}\ |x_k| < +\infty \}$ equipped with the norm $\| x\| = \mathrm{sup}_{k\in\mathbb{N}} |x_k|$ is NOT a Banach ...
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1answer
304 views

Basic question on bounded linear operators in Banach spaces.

I have sincerely tried this problem, for way too long, and I must admit defeat. How am I to prove the following? Let X be a Banach space and I be the identity mapping on X. If T is a bounded linear ...
2
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1answer
102 views

metric projection onto one dimensional (closed ) subspace in $L_p (p\neq 2)$

I want to know "if the metric projection onto one dimensional (closed) subspace in $L_p (p\neq 2)$ is linear? I think it is not linear, but I can not give a strict proof. Thanks for any answer!
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155 views

Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$

Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$. I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two ...
3
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117 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
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1answer
272 views

How to show that this is a complete metric space [duplicate]

Let $(X;d)$ be a metric space and $\mathrm{C_b}(X,\mathbb{R})$ denote the set of all continuous bounded real valued functions defined on $X$, equipped with the uniform metric: $$ \rho(f,g) = \sup\{\, ...
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1answer
98 views

$L_{k}^{1}([0,1])$ is a Banach space

Let $L_{k}^{1}([0,1])$ be the space of all $f\in C^{k-1}([0,1])$ such that $f^{(k-1)}$ is absolutely continuous on $[0,1]$ (and hence $f^{(k)}$ exists a.e. and is in $L^{1}([0,1])$). Show that ...
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3answers
58 views

generalization of a normed space

I study analysis and have a problem: I have a normed space for example $(X,M)$ that is not complete, how can I complete the space $X$ with respect to norm $M$? please help me Thanks
3
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1answer
661 views

Banach spaces and their unit sphere

Let $X$ be a normed vector space. Show that if a subsequence of a Cauchy sequence converges, then the whole sequence converges. Use the part 1 to show that $S = \{x\in X : \|x\| = 1\}$ is complete ...