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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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is this set invariant under a operator?

Let $E$ be a Banach space, and $T:E\rightarrow E$ a continuous bounded mapping. Let $x_0\in E$ and $x_n=T(x_{n-1})$, $C=\overline{conv}(x_0,x_1,...,x_n,...)$. Is $U$ invariant under the operator $T$...
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1answer
26 views

Banach spaces $X,Y$ and a mapping $f:X\to Y$

Let $X,Y$ be Banach spaces and a mapping $f\in \mathcal{L}(X,Y)$. Suppose that $f$ is also an injective map and an open map from $X$ to $Y$. Show that then $f \in \mathcal{B}(X,Y).$ Here I denoted ...
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1answer
43 views

Prove or disprove that $W$ is a Banach space.

True or false, and justify. Let $W =\{ P: \textrm{polynomial of degree} \le100\}$ and $|| P || = \sum|a_n|$. Then $|| P ||$ is a norm and $W$ is a Banach space under this norm. I proved that ...
2
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1answer
24 views

Dual spaces of uniformly convex Banach spaces

I am interested in the dual spaces of uniformly convex Banach spaces. Given a uniformly convex Banach space $X$, can anything be said about uniform convexity of its dual space $X^*$? Or given a ...
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2answers
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Limit of non-degenerate biliniear forms

Let $X$ be a Banach space (in my specific case I have $X=C_b(\mathbb{R})$) and let $\{B_n\}_{n\geq 1}$ be a sequence of biliniear forms $B_n:X\times X\rightarrow \mathbb{C}$, which are non-degenerate ...
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1answer
45 views

Space of sequences such thtat $\sum_{n=0}^{\infty}2^n|a_n|<+\infty$

Consider the space of sequences of real numbers $\{\vec{a_i}\}$ where each $\vec{a_i}=\{a_{i_n}\}$ is such that $\sum_{n=0}^{\infty}2^n|a_{i_n}|<+\infty$. Then how could we better describe the ...
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2answers
81 views

$A_1,A_2$ fulfill property, but their sum $A_1+A_2$ does not

Let $V$ be a real, reflexive, separable Banach space. Are there operators $A_1,A_2: V \to V^*$ that fulfill the property \begin{cases} u_n \rightharpoonup u \\ A_iu_n \rightharpoonup b \\ ...
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+50

If X has beta property, do $(x_i)_i$ points are LUR?

Let $X$ be a Banach space, $S_X$ its unit sphere and $B_X$ its unit ball. The space $X$ is said to has $\beta$ property if there exists a system $\{(x_i,f_i):i \in I\} \subset S_X \times S_{X^*}$ and ...
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compactness and relative compactness

Suppose that $M$ is compact subset of a Banach space X. Is $M$ relatively compact too? As far as I know, there is a characterisation of compact sets via Hausdorff $\varepsilon$ - net theorem and it'...
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1answer
29 views

Norm of a bounded linear functional.

Let $X=(\mathbb R^2, \|.\|_3)$ be a real normed space, where $\|(x_1,x_2)\|_3=[|x_1|^3+|x_2|^3]^{1/3}$. How to find the norm of bounded linear functional $ax+by$? I tried this way: $|ax+by|\leq |a||x|...
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Uniform convexity requires stronger triangle inequality to be true uniformly.

From the wikipedia (in the section "Properties") https://en.wikipedia.org/wiki/Uniformly_convex_space : The strict convexity means a stronger triangle inequality $\|x+y\|<\|x\|+\|y\|$ holds ...
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1answer
48 views

A counterexample of Banach Steinhaus Theorem

I was reading about a consequence of Banach-Steinhaus theorem which states that: Let $E$ be a Banach space and $F$ be a normed space, and let $\{T_n\}_{n\in \mathbb{N}}$ be a sequence of bounded ...
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1answer
27 views

Proving that if the dual space $X^*$ of a Banach space $X$ is separable , then $X$ is separable

I have reading an answer post before , and there's something I do not understand . For each $f_n$ in the unit ball of $X^*$ , why there exist $x_n \in X$ with $\|x_n \| \le1$ such that $f_n(x_n) \ge \...
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1answer
18 views

A linear functional $l$ on a Banach space $B$ is continuous if and only if $\{f \in B : l(f)=0 \}$ is closed .

Suppose $B$ is a Banach space and $S$ a closed proper subspace , and assume $f_0 \notin S$ . (a) Show that there is a continuous linear functional $l$ on $B$ , so that $l(f)=0$ for $f \in S$ , and $...
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1answer
25 views

The subspace $S$ defines an equivalence relation $f \sim g$ to mean $f-g \in S$. Show that $B/S$ is a Banach space

Suppose $B$ is a Banach space and $S$ is a closed linear subspace of $B$ . The subspace $S$ defines an equivalence relation $f \sim g$ to mean $f-g \in S$ . If $B/S$ denotes the collection of these ...
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79 views

Closed subspace and quotient norm

In the Banach space $C[0,1]$ consider the subspace $M=\lbrace g \in C[0,1]: \int_{0}^{1}g(t)dt=0 \rbrace $ Show that $M$ is closed in $C[0,1]$ and calculate the quotient norm $(\|f+M \|)$ where $f(t)...
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1answer
36 views

Closed and finite subspace [duplicate]

Given $X$ a normed space, $M$ a closed subspace of $X$ and $Z$ a finite dimension subspace of $X.$ Show that $M+Z$ is a closed subspace of $X.$ How can I do this? I thought to construct a sequence ...
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Applications of $L_p$ spaces

I'm studying Lebesgue integration theory and understand the definition of $L_p$ spaces. What can we do with $L_p$ spaces?
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1answer
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Banach space and normed space [closed]

Can someone please help me solve this statement? Consider the vector space $C^1 [0, 1]$ of the differentiable functions with continuous derivative in the interval $[0, 1]$. For $f\in C^1[0,1]$, let $...
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1answer
67 views

Convergence of a sequence on the unit sphere of Bahach or Hilbert space

Let $X$ be a Banach or Hilbert space and $A$ be a bounded linear operator on $X$, and fix an element $x \in X$. Then I want to know that are there any good ways or theories to deal with the ...
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39 views

Set of injective operators is a dense residual set in $\mathcal{B}(\mathfrak{X})$

Let $\mathfrak{X}$ be a Banach space and $\mathcal{B}(\mathfrak{X})$ the set of bounded linear operators mapping $\mathfrak{X}$ to $\mathfrak{X}$. In [1] below it is shown that the set of invertible ...
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Weak closure of subsets of the unitary sphere of a Banach space.

Assume that $(X,\|\cdot\|)$ is a Banach space with $\|\cdot\|$ strictly convex. Define $S=\{x\in X:\|x\|=1\}$. Suppose that $\varepsilon>0$ and $x_0\in S$ and define $$ B_\varepsilon=\{x\in X:\|x-...
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Question about smoothness of a Banach space.

Define $\delta: [0,2] \to [0,1]$ defined by $$\delta_U(\epsilon) = \inf \bigg\{\frac{1}{2}\bigg(2-\|u_1+u_2\|\bigg): \ u_1, u_2 \in U^0, \|u_1-u_2\| \geq \epsilon\bigg\},$$ where $U^0$ is the ...
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0answers
9 views

Unconditional constant vs Suppression unconditional constant

I am trying to solve the following question. Let $(u_n)_{n=1}^{\infty}$ be an unconditional basis for a Banach space $X$ with suppression-unconditional constant $K_{su}$. Prove that for all $N$, ...
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0answers
36 views

Properties of Hilbert space valued functions

Let $f:X\to Y$ be a Hilbert space valued function, with $Y$ a separable Hilbert space and $X$ a measurable space. Furthermore, assume that $$ \int \|f(x)\|_Y \mu(dx)<\infty, $$ for $\mu$ is a ...
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2answers
209 views

Showing a function is Frechet Differentiable?

I just started learning the Frechet Derivatives. So I have a function $H:\mathbb{R}^{N\times n}\to\mathbb{R}^{N\times n}$, i.e. $U^T\in\mathbb{R}^{N\times n}$ and $$H(U^T)=GW\times (F(U))^T+S\times U^...
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1answer
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Hyperbolic tangents as a dense subset of smooth functions satisfying certain conditions

Edit: The original question contained some errors that lead to some comments and answers that do not apply anymore. I thank those that took time to read the question and pointed out the errors. I ...
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0answers
73 views

Optimization in Banach space: Find functions that minimize the supremum of a convex operator.

Let $D \subset \mathbb{R}^n$ be compact. Denote by $C(D, \mathbb{R}^n)$ the space of continuous functions from $D$ to $\mathbb{R}^n$. Let $K$ be a real, symmetric, positive-definite $n \times n$ ...
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82 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 ...
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Reason for the term “smooth”

A normed space $X$ is said to be smooth if for $x \in X$ with $||x||=1$ there exists a unique bounded linear functional $f$ such that $||f||=1$ and $f(x)=||x||$. Why the term "smooth" comes?
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Suppression-unconditional constant can replace the unconditional constant

I am trying to solve the following question. Let $(u_n)_{n=1}^{\infty}$ be an unconditional basis for a Banach space $X$ with suppression-unconditional constant $K_{su}$. Prove that for all $N$, ...
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39 views

Topics in Banach space theory

Let $X$ be a subspace of a space with unconditional basis. Show that if $X$ contains no copy of $c_0$ or $l_1$ then $X$ is reflexive. We know that a Banach space is reflexive iff it has a basis which ...
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if f is linearly bounded in its second argument then each solution to u'(t)=f(t,u(t)) is bounded.

Let $(X,\Vert \Vert)$ be a Banach space and $f:\mathbb{R} \times X \rightarrow X$ continuous. If f is linearly bounded in its second argument, i.e. there exist $\alpha, \beta \in C(\mathbb{R} ; \...
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2answers
19 views

Is $C^1([a,b], \mathbb{R}^n)$ a reflexive Banach space?

I want to prove or disprove that $C^1([a,b], \mathbb{R}^n)$ equipped with the norm $||x||=\underset{t\in[a,b]}{\sup}|x(t)|_{\mathbb{R}^n}+\underset{t\in[a,b]}{\sup}|\dot{x}(t)|_{\mathbb{R}^n}$ is a ...
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1answer
82 views

The last digit of pi (in terms of Banach limits)

Let $\phi : l^\infty \to \mathbb C$ be a Banach limit, and define the sequence $\{x_k\}_{k\geq 0}$ to be the digits in the 10-base decimal expansion of $\pi$. Note that $$\{x_k\}_{k\geq 0} \in l^\...
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34 views

Show that C [a,b] is a Banach space

Let $C [a,b]$ be a set of all real-valued functions $x(t),y(t),...$ which are functions of an independent real variable $t$ and are defined and continuous on a closed interval $[a,b]$. Show that $C [a,...
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Relation between two arbitrary vectors in a convex cone in a Banach space.

Let $x,y$ be two arbitrary vectors in a convex cone in a Banach space. Does there exist some relation between these two vectors?
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Geometry of Banach Space (Radon Nikodym)

I read in David Williams, Probability with martingale : "The right context for appreciating the close inter-relations between martingale convergence, conditional expectation, the Radon-Nikodym theorem,...
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1answer
29 views

Isometries on a Banach space converging pointwise

I'm trying to find a Banach space $V$ with closed unit ball $B$ and a sequence of isometries $(f_n:V\to V)$ such that $(f_n)$ converges pointwise in $B$ but not uniformly in $B$. My first attempts ...
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2answers
23 views

Banach spaces, $C^1([-1,1])$

Denote by $C^1([a,b])$ the vector space of continuously differentiable functions with derivatives at $a,b$ defined as one-sided limits. I was wondering whether the linear subspace $C^1([-1,1])$ of $C^...
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0answers
26 views

The whole sequence of a convergent subsequence

Let $(E,\left \| . \right \|,\sqsubseteq )$ be a partially ordered Banach space. Let $(x_n)_{n\in\mathbb N}$ a monotone sequence (let's say $u_n\leq u_{n+1}$), such that ${u_n}$ has a convergent ...
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2answers
57 views

$\operatorname{Ker}(B) \subset \operatorname{Ker(A)}$ if and only if ${\left\| {Ax} \right\|_X} \leqslant \alpha {\left\| {Bx} \right\|_X}$

Is this statement is correct: Let $X$ be a Banach space, and let $A$ and $B$ two continuous operators on $X$ , do we have for some constant $\alpha$ the following $${\left\| {Ax} \right\|_X} \...
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1answer
29 views

How to show that if $X=M \oplus N$ is a Banach space then $\exists c > 0$ $\forall m\in M \, \forall n\in N: ||m||+||n|| \leq c ||m+n||$? [closed]

Let $X=M \oplus N$, where $X$ is a Banach space and $M$ and $N$ are closed subspaces of $X$. How to prove that $\exists c > 0$ constant such that $\forall m \in M$ and $\forall n \in N$ $$ \left\...
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1answer
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Proving that $Y= \{x_n \in \ell^2 : x_{2n} =0, n \in \mathbb N\}$ is closed and finding $Y^\bot$.

Exercise : Show that the space $$Y= \{x_n \in \ell^2 : x_{2n} =0, n \in \mathbb N\}$$ is a closed subspace of $\ell^2$ and find the space $Y^\bot$. Attempt-Thoughts : First of all, the $\ell^...
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1answer
25 views

Set of all sequences converging to $0$ forms a Banach Space.

Let $c_0 =\{(x_n)\in \mathbb{R}:x_{n}\to 0\},$ then show that $c_0\subset l^{\infty}$ and that $c_0$ forms a Banach Space. Since convergent sequences are bounded clearly $c_0\subset l^{\infty}.$ ...
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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 ...
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0answers
9 views

$(V, \geq)$ Banach lattice, $(W, \geq)$ Riesz space, then $\mathcal{L}(V, W)^+ = \mathcal{B}(V, W)^+$

Let $(V, \geq)$ be a Banach lattice and $(W, \geq)$ be Riesz space whose whose positive cone is generated by some positive element of the space. Then $\mathcal{L}(V, W)^+ = \mathcal{B}(V, W)^+$, where ...
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1answer
30 views

A Banach space with a cone's order

I'm reading a paper on a Banach space with an order induced by a cone. Let $(\mathbb{X}, || \cdot||)$ be a real Banach space. We define a subset $P$ of $\mathbb{X}$ by $P := \{ x \in \mathbb{X} : x\...
8
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1answer
72 views

Dense subset of two Banach spaces also dense in the intersection

My question is: Let $ V $ be a vector space (over $ \mathbb K\in\{\mathbb{R}, \mathbb{C}\} $), $ X,Y\subseteq V $ two subspaces equipped with norms $ \|\cdot\|_X, \|\cdot\|_Y $ such that $ (X,\|\cdot\...
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1answer
35 views

Banach space with subset whose elements are at least $d\gt 0$ far from each other is not separable

Let $X$ be a Banach space, and $A\subseteq X$ subgroup, where $A$ is not countable, and there is some $d \gt 0$ such that for all $x,y \in A$: $||x-y||>d$. Prove that $X$ is not separable. My ...