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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.

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Show that $L_{p,\infty}(0,\infty)/L_{p,\infty}^0(0,\infty)$ is a reflexive space

Let $L_{p,\infty}(0,\infty)$, $1<p<\infty$, be the weak $L_p$-space on $(0,\infty)$ and $L_{p,\infty}^0(0,\infty)$ be its separable part (the closure of all bounded functions having finite ...
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0answers
32 views

$\ell_1$-Norm Minimization

I am trying to solve the following optimal control problem: $$ \min_{u\in\ell_1}\|u\|_1 \qquad \text{subject to} \qquad Cu = x_f $$ with the dynamics $$ x_{k+1} = Ax_k+Bu_k $$ This is a discrete ...
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2answers
25 views

Is $X$ reflexive if $X=\cup Y_i $ and $Y_i$ is reflexive for every $i$?

Let $X$ be a Banach space. Assume that there are a family of closed (with respect to $\left\|\cdot\right\|_X$) subspaces $Y_i$ of $X$ such $X=\cup Y_i $ and $Y_i$ is reflexive for every $i$. Can we ...
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1answer
43 views

Uniqueness of Hahn-Banach extensions for $c_0 \subseteq \ell^\infty$

Let $c_0$ be the space of real sequences converging to zero with supremum norm. $c_0$ is a (closed) subspace of $\ell^\infty$, the space of bounded real sequences. A $f \in {c_0}^*$ corresponds to a $...
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1answer
20 views

Convergence of a sequence by the convergence of a subsequence

Let $X$ be a Banach space and $\{x_{n}\}_{n\in\mathbb{N}}$ be a bounded sequence in $X$. Assume for any subsequence $\{x_{n_{k}}\}_{k\in\mathbb{N}}$ of $\{x_{n}\}_{n\in\mathbb{N}}$, there exists a ...
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0answers
67 views

Does $g_n \rightarrow 0$ converge weakly?

This is where I am stuck while solving another problem. Let $T:L^1 \rightarrow X$ be an operator such that $T|_{L^2(\mu)}$ is compact. Suppose $f_n$ be a sequence in $L^1$ such that $f_n \...
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0answers
23 views

Semivariation of a vector measure in dual space

We define the semivariation of a vector measure $\mu:\mathcal{B}(\mathbb{T})\to E$ as $$\|{\mu}\|(A)=\sup\lbrace{|\langle e^*,\mu\rangle|(A),\; e^*\in E^*,\;\|e^*\|=1\rbrace}$$ where $\langle e^*,\...
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1answer
22 views

Prove that $X_0$ is a closed subspace of $X$

I encounter the following exercise in functional analysis: Let $(X,\|\cdot\|_{X})$ and $(Y,\|\cdot\|_{Y})$ be Banach spaces and $\{T_n\}$ be a family of uniformly bounded linear maps form $X$ to $Y$, ...
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1answer
18 views

convergence for the weak-* topology

Let E be a Banach space. Let $(x^∗_n )$ be a sequence in $E^∗$ verifying $(<x^∗_n , x>)$ converges for any $x ∈ E$. Prove that $\exists x^∗ ∈ E^∗: (x^∗_n )$ converges vers $x^*$ for the weak-∗...
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95 views

When $|a-b|=|a|-|b|$?

Could someonoe help me to decide if the following satetement is true? If $K$ is a strictly convex Banach space and $a,b\in{K}$ verify $|a-b|=|a|-|b|=1$ then, $a=\lambda{b}$ for some $\lambda\geq{0}$. ...
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0answers
46 views

Every operator $T : X \rightarrow L^1$ is weakly compact.

I am trying the following problem. Let $X$ be a Banach space that does not contain a copy of $l^1$. Show that every operator $T : X \rightarrow L^1$ is weakly compact. I am not sure how to start. ...
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0answers
30 views

dimension of closure of vector subspace

Let $F \subseteq H$ be a subspace of a Hilbert (or Banach, or normed) space. is it true that$$\dim(F) = \dim (\bar{F})$$ where we are working with Hilbert dimension.
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1answer
37 views

Banach space inequality

I'm looking to prove the following inequality \begin{align} ||\frac{u}{||u||}-\frac{v}{||v||}|| \leq 2||u-v|| \end{align} where $u$ and $v$ are elements of a Banach space such that $||u||$ and $||v|...
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62 views
+50

Dunford Pettis property

I am trying the following problem. Let $\mu$ be a probability measure. Show that an operator $T : L_1(\mu) \rightarrow X$ is Dunford Pettis if and only if $T|_{L_2(\mu)}$ is compact. I was trying to ...
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1answer
26 views

Convexity is crucial in Schauder-Tychnonov fixed-point theorem.

The following Theorem is well-known: Schauder-Tychnonov fixed-point theorem: Let $K$ be a compact convex subset of a Banach space, $E$. If $T:K\to K$ is continuous, then $T$ has a fixed point. I'm ...
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0answers
38 views

The map $f:\bar{B}\to \bar{B}, $ continuous such that $ \bar{B}\subseteq C[0,1],$ unit closed, need not have a fixed point.

The map $f:\bar{B}\to \bar{B}, $ continuous such that $ \bar{B}\subseteq C[0,1]$, need not have a fixed point. Know about the Brouwer fixed point Theorem on $\mathbb{R} ^n$ which states that if $ \...
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0answers
26 views

An example of differential equation in Banach space with no solution

Consider the following problem $$x'(t)=f(t,x(t)),\qquad x(t_0)=x_0.$$ If $Q_b=\{(t,x)\in\mathbb{R}\times X\,|\,|t-t_0|\le a\, \|x-x_0\|\le b\}$ and function $f\colon Q_b\to X$ is continuous and ...
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1answer
12 views

When can we replace countably valued simple functions by finitely valued simple functions

Suppose that $(\Omega,\mathcal{A},\mu)$ is a finite measure space and $X$ is a Banach space. Let $f:\Omega \to X$ be a function that is an a.e. pointwise limit of countable-valued functions $f_n:\...
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1answer
21 views

Denseness of Banach space in its double dual [closed]

Let $X$ be a Banach space, $X^{''}$ its continuous double dual and $\phi$ the cannonical embedding $X\to X^{''}$. Is $\phi(X)$ always dense in $X^{''}$?
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1answer
24 views

Hahn-Banach and the Fundamental Theorem of Calculus for Banach-space valued functions

I am trying to understand the proof of the Fundamental Theorem of Calculus for Banach space-valued functions, and in particular, how the Theorem of Hahn-Banach is being applied there. In the ...
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0answers
30 views

Immersion duals spaces

Is correct say that $(\ell_\infty)'\subset (c_0)'$? My idea is the following: I know that $c_0\subset\ell_\infty$. Then, the function $$ \begin{split} \Psi:(\ell_\infty)'&\to(c_0)'\\ f&\...
2
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1answer
124 views

Banach space and Hilbert space topology

Let $B$ be a Banach space. It is not necessarily true that there exists a Hilbert space $H$ linearly isometric to $B$. However, is it true that there exists a Hilbert space $H$ homeomorphic to $B$?
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2answers
80 views

If $C(K)$ is isomorphic to $C(H)$, are $K$ and $H$ homeomorphic?

Let $K,H$ be compact Hausdorff spaces. The Banach Stone theorem says that $K$ and $H$ are homeomorphic $\iff$ $C(K)$ and $C(H)$ are isometrically isomorphic. Is it true that if $C(K)$ and $C(H)$ are ...
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1answer
52 views

How does one show that $\{b_n\}^∞_{n=1}$ does not not converge in $c_o$

Let $X = \prod_{k=1}^\infty \{0,1\}$ Given that for each $k$ = $1, 2, 3$, . . . the projection onto the $k^{th}$ coordinate $π^k$ : X → {0, 1}, given by $π^k$($\{x_n\}^∞_{n=1}$) is Lipschitz If $\{...
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0answers
20 views

$(C(A,\mathbb{R}),\|\cdot\|_\infty)$ is a Banach Space when $A$ is Compact

For a normed vector space $(V,\|\cdot\|_{V})$, let $A\subseteq V$ be compact, and let $(C(A,\mathbb{R}),\|\cdot\|_{\infty})$ be the space of continuous functions from $A$ to $\mathbb{R}$ with respect ...
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0answers
44 views

Question: $X \hookrightarrow Y$ implies $ Z \cap X \hookrightarrow Z \cap Y$

Let $X,Y,Z$ be some Banach spaces and assume that $X \hookrightarrow Y$, i.e $X$ is embedded continuously into $Y$. Can we claim then that $ Z \cap X \hookrightarrow Z \cap Y$? By a set theory ...
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1answer
12 views

$C[X,R]$ a closed subspace of $B[X,R]$

Is there any condition for $X$ to be a compact space in the theorem "Space $C[X,R]$ is a closed subspace of $B[X,R]$? Because I see that if $X$ is not compact then $C[X,R]$ does not become a subset of ...
1
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1answer
26 views

Variant of the Contraction Mapping Theorem

Let (C, ||.||) be a closed subset of a Banach space, and let f: C -> C be a mapping such that ||f(x) - f(y)|| < ||x - y|| for all x, y in C. Must there exist a fixed point in C that f maps to ...
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0answers
127 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 $...
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1answer
59 views

functional analysis : to show this space is a Banach space [on hold]

Let $1\leq p <\infty$ and $n\geq 1$ and let $W_p^n[0, 1] = $ the functions $f_[0, 1]\to \Bbb F$ such that $f$ has $n-1$ continuous derivatives, $f^{(n-1)}$ is absolutely continsous, and $f^{(n)}\in ...
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1answer
219 views

Is a linearly independent set whose span is dense a Schauder basis?

If $X$ is a Banach space, then a Schauder basis of $X$ is a subset $B$ of $X$ such that every element of $X$ can be written uniquely as an infinite linear combination of elements of $B$. My question ...
2
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1answer
55 views

Showing linear, continuous operator from Banach space into particular quotient space is open

Let $X$ be a Banach space, let $Y$ be a closed subspace in $X.$ Let a linear, continuous operator $\pi \colon X \to X/Y$ be defined by $\pi(x) = \overline{x}.$ I'd like to show $\pi$ open. $\...
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0answers
18 views

Reference Request: Projection from $L^1$ onto $L^2$

Suppose that $(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space. Since $L^2(\Omega,\mathcal{F},\mathbb{P})$ is a subspace of $L^1(\Omega,\mathcal{F},\mathbb{P})$, is there a well-...
2
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1answer
78 views

Let $A: X \rightarrow Y$ a bijective continous map between two Banach spaces X and Y. Then, $A^{-1}$ is also continous?

We know if A is a map continuous, bijective and linear then the answer is yes, $A^{-1}$ is continuous. But, if $A$ is no linear then $A^{-1}$ is also continuous ?
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0answers
63 views

Closed hyperplanes on a normed space are isomorphic

Let $X$ be a normed space. I'd like to show that all closed hyperplanes in $X$ are isomorphic. My attempt Let $H$ and $W$ be closed hyperplanes. We know $\dim(X/H)=\dim(X/W)=1$, therefore $(X/H)$ ...
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1answer
19 views

$X$ Banach, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$ implie that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$.

Exercise : Let $X$ be a Banach space, $u_n \to u$ and $x^*_n \xrightarrow{w^*} x^*$. Show that $\langle x^*_n, u_n\rangle \to \langle x^*,u\rangle$. Attempt-Discussion : I know that a sequence $...
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1answer
40 views

If $A+B$ is a compact non-zero operator, which properties follow for $A$ and $B$?

Assume we are in $l^p$ spaces or at least Banach spaces. I'm trying to find out what the knowledge that the linear operator $A+B$ is compact tells me about the properties of the linear operators $A$ ...
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1answer
41 views

Mean Value Inequality in Banach Space without Hahn-Banach or Integrals

If $f : E \to F$ is a continuous map of Banach spaces, with bounded Fréchet derivatve. Then $x_0,x_1 \in E\Rightarrow \|f(x_1) − f(x_0)\| ≤ M\|x_1 − x_0\|$ where $M = \sup \|f'(x)\|.$ The most ...
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0answers
24 views

Norm in $ \ell_{\infty} /c_0 $ [duplicate]

Show that norm in $ \ell_{\infty} /c_0 $ is $||\overline{x} ||= \text{lim sup}|x_i|$ for each $x=(x_i)_i \in \ell_{\infty} $. I'm doing $$||x_n+c_0||= \text{inf}_{y_n \in c_0}||(x_n)+(y_n)||_{\infty}=...
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0answers
28 views

Bochner integral on non complete space

Let $X_0=(X,\|\cdot\|_0)$ be a Banach space and $X_1=(X,\|\cdot\|_1)$ a normed vector space (not complete) such that there is a constant $c>0$ such that \begin{align*} \|\cdot\|_1 \leq c\|\cdot\|_0 ...
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1answer
37 views

Does closed convex sets having unique nearest points imply the parallelogram law?

It's a well-known result that if $X$ is a Hilbert space, then for any closed convex subset $C$ of $X$, there exists a unique element of $C$ with minimal norm. I'm wondering whether the converse is ...
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1answer
31 views

Hilbert space with two equivalent norms

Let $H$ be a Hilbert space with norm $\|\cdot\|_1$ induced by the inner product. And if we define another norm on $H$, we will denote this norm $\|\cdot\|_2$. Now we suppose that the two norms $\|\...
1
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1answer
63 views

On weak*-sequentially completeness

I want to prove that every dual space is weak*-sequentially complete. Let $X$ be a normed linear space and let $(f_n)$ be a weak* Cauchy sequence in $X^*$. Thus for all $x\in X$, $(f_n(x))$ is a ...
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1answer
28 views

Isometric embedding of $L^2$ onto $H^{-1}$

Let $X$ be a Banach space. Many sources in the literature identify $L^2(X)$ with $H^{-1}(X)$ through the identification $$ \varphi: L^2(X) \to H^{-1}(X); \quad \quad \varphi(u)(v) := (u,v)_{L^2}, \...
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0answers
11 views

Constant of continuity as a monotonically increasing function of the radius of the neighbourhood

While reading the fairly famous paper by Brezzi et al from 1980 (link), I got slightly puzzled by one of the conditions, in particular in Theorem 1 (Implicit Function Theorem basically). They consider ...
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0answers
34 views

topologically complemented subspace

Let $ X$ be a Banach space and let $ M $ be a topologically complemented subspace of $ X$. If a closed subspace $ N \subset X$ is isomorphic to $ M $, is it topologically complemented too?
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1answer
27 views

Absolutely continuous Banach space valued function

Let $X$ be a Banach space and $F:[a,b] \to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f \in L^1([a,b],X)$, is the ...
1
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1answer
38 views

If operator is closed and densely defined then $D(A^*)^\perp = \{0\}$

I'm a bit rusty in my Functional Analysis and couldn't solve this question: Let $X$ be a Banach space (over either $\mathbb{R}$ or $\mathbb{C}$) and $X^*$ its dual space. Show that, if $A:D(A) \...
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1answer
53 views

Prove $\|T\| = \sup_{\|x\| < 1} \|Tx\|$

Let $X, Y$ be Banach spaces. And $T \in B(X\rightarrow Y)$. Prove that $$\|T\| = \sup_{\|x\| < 1} \|Tx\|$$ Discussion Having trouble seeing how to handle some of these ideas below. Please let me ...
0
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1answer
11 views

Confirmation of convergence in subdomain

Let $\{u_{n}\}_{n\in\mathbb{N}} \subset L^{p}(\Omega)$ be a sequence of function such that $u_{n} \to u$ in $L^{p}(\Omega)$ for $\Omega\subset \mathbb{R}^{N}$ bounded domain. I want to show that for ...