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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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$L^2$ and $C^0$ norms combined

We know that $L^2$ and $C^0$ are complete metric spaces. Now consider for fixed $T$ the space of real-valued processes $X:\Omega\times [0,T] \rightarrow \mathbb R$ (with corresponding $\sigma$-...
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Every bounded net in a dual space has a cluster point?

Question: Let $X$ be a normed space and $X^*$ be its continuous dual of $X.$ Assume that $(x_\alpha^*)_\alpha$ is a bounded net in $X^*.$ Is it true that there exists a cluster point $x^*$ in $X^*$ ...
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Show that the following is bounded and surjective (Folland exercise 5.36) [duplicate]

Let $X$ be a separable Banach space and let $\mu$ be the counting measure on $\mathbb{N}$. If $\{x_n\}_{n=1}^{\infty}$ is a countable dense subset of the unit ball of $X$, and $T:L^1(\mu)\to X$ is ...
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Tensor product of function spaces: does $\widehat{\bigotimes}_{\pi;1\leq k\leq n}C(X_{k})^{\prime}\cong C(\prod_{k}X)^{\prime}$ hold?

Let $n\in\mathbb{N}$ with $n\geq 2$. Let $X_{k}$ be (non-empty) compact Hausdorff spaces for $1\leq k\leq n$. Let $X:=\prod_{k}X_{k}$ and let $\pi_{k}:X\longrightarrow$ be the project functions. It ...
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1answer
37 views

Normed space $C^2[0,1]$ with norm $\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$ is Banach space

The problem is as follows: I want to show that the normed space $C^2[0,1]$ with norm defined as $$\lVert f\rVert:=\max_{t\in[0,1]}\{\lvert f(t)\rvert+\lvert f''(t)\rvert\}$$ is a Banach space (and I ...
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Showing weakly continuous operators are continuous without using weak topology

Let $X$ and $Y$ be Banach spaces, and let $T:X\rightarrow Y$ be a linear map such that $f\circ T$ is continuous for all $f\in Y'$. Show that $T$ is continuous. Now I think this problem is trivial ...
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Cardinal number of the Banach space $c_0(A)$

For infinite dimensional Banach space $X$, the cardinality of $X$ is equal to its algebraic dimension. Now let $A$ be an infinite set and define $c_0(A)=\{f:A\rightarrow\Bbb R, f$ is bounded and $ \...
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1answer
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Are neural networks with bounded parameters a compact subset of the Banach space of continuous functions?

Let $d, n \in \mathbb{N}$. Moreover, let $D \subset \mathbb{R}^d$ be compact and denote with $\mathcal{C}(D, \mathbb{R}^n) $ the set of continuous functions from $D$ to $\mathbb{R}^n$. Then $\mathcal{...
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Is this mapping from a hypercube into a Banach space of continuous functions itself continuous?

Let $d, n \in \mathbb{N}$. Moreover, let $D \subset \mathbb{R}^d$ be compact and denote with $\mathcal{C}(D, \mathbb{R}^n) $ the set of continuous functions from $D$ to $\mathbb{R}^n$. Then $\mathcal{...
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Proving that $\mathcal{C}([0,1])$ is a Banach space

Let $V=\mathcal{C}([0,1])$ be the space of all complex valued continuous functions with the norm $||f||_{\infty}=\sup_{x\in[0,1]}|f(x)|$. Then, $V$ with the norm $||\cdot||_{\infty}$ is a Banach space....
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Weak Convergence Lemma - Is Banach needed?

Lemma. Let $X$ be a normed space. If $x_n \rightharpoonup x$ in $X$ and $x_n^* \to x^*$ in $X^*$, then $\lim_{n \to \infty} x_n^*(x_n) = x^*(x)$. If $X$ is even Banach, then $x_n \to x$ in $X$ and $...
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isometric isomorphism of Banach spaces

Let $X$ and $Y$ be two Banach spaces and $Z$ be a dense subspace of $X$, which is not closed. Could anyone help me to show that $\phi :B(X,Y)\rightarrow B(Z,Y)$ s.t $\phi (T)=T|_Z$ is isometric ...
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Stronger convergence of Markov chains and non-dissipative property

Let $P$ be the transition probability matrix of a Markov chain on $\mathbb{N}$. Then $P$ acts as a bounded operator on $\ell^1$ by $\mu \mapsto \mu P$ for $\mu \in \ell^1$. Identify $\ell^1$ with the ...
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the spectrum of a bounded linear operator on $X\times X$

If we consider $X\neq\{0\}$ to be a complex Banach space then the product $X\times X$ is a Banach space with the norm $\|(x,y)\|=\|x\|+\|y\|$. $T(x,y)=(x + y,x - y)$ is then a bounded linear operator ...
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The spaces $\ell^p, \; 1 \leq p < + \infty$ are separable. On the other side, $\ell^\infty$ is not.

Exercise : Show that the spaces $\ell^p, \; 1 \leq p < + \infty$ are separable. Attempt : In order to show that $\ell^p$ is separable for $\ell^p, \; 1 \leq p < + \infty$, we need to work ...
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Conclusion about Banach space

Let $X,Y$ be normed spaces and $f \in X^*$ a linear functional with $||f||=1$. Let $B(X,Y)$ be the set of all bounded linear transformations from $X$ to $Y$. Define $T:Y \to B(X,Y)$ by $(Ty)(x) = ...
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Banach Spaces, Convergence and Spectrum

Let $V$ be a Banach space and $T_n → T$ in $B(V)$. Assume $λ_n ∈ σ(T_n)$ and $λ_n → λ$, I want to show that $λ ∈ σ(T)$. Okay, so if $\lVert T_n-T\rVert_{\mathcal B(V)}\to 0$ and $\lambda_n\to \...
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Covering from Dense Projection

Let $V$ be a Banach space, $W$ a (strict) subspace of $V$, and $U$ a dense proper subset of $V$. When is does there exist a (linear) projection $P^V_W:V\twoheadrightarrow W$, such that $$ P_W^V(U)=W ....
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1answer
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Need Hint; Show that the Limit Exists when $f\in C^1(0,1)$ and…

The problem is as follows: Assume that $f\in C^1(0,1)$ and $$ \int_{(0,1)}x|f'|^p\,dx<+\infty\qquad\text{for some }p>2. $$ Show that $\lim_{x\rightarrow 0^+}f(x)$ exists. Note: $C^1(0,1)$...
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3answers
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Motivation: Open Mapping Theorem

One of the Fundamental Theorems in Functional Analysis is the Open Mapping Theorem. Theorem. Let $X,Y$ be Banach spaces and $T \in L(X,Y)$. Then $T$ is surjective if and only if it is open. I'd love ...
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surjectivity of an operator which is defined on some set of analytic functions

For a fixed function $h\in H(\mathbb{D})$ and a fixed complex number $λ$ let $f$ (It is analytic) be a solution of $$(λI − C)f = h.$$ Using Taylor expansions, it is easy to see that the operator $(λI −...
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How to use transfinite induction/recursion to construct a dense Hamel Basis of a Banach space?

Here is an excellent answer by David C. Ullrich to my old question. In his answer, he proves the following theorem by doing a transfinite induction on the cardinality of the base of topology. If $X$...
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1answer
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From $L^p$ to $L^q$ Norm for finite trigonometric polynomial

Let $1\leq p\leq q \leq \infty$ and let $f\in L^p([0,2\pi])$ such that $\hat{f}(k)=0$ for all $\vert k\vert > N$. I would like to show that $$\Vert f \Vert_p \leq (2N-1)^{1/p-1/q} \Vert f \Vert_q....
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1answer
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Do weak convergence and convergence of norms imply convergence in $L^1$?

I know this does hold in $L^2$, since it's a Hilbert space. I suspect that this is not true, but I cannot think of a counterexample. Specifically, I want to know if $f_n \xrightarrow{w} f$ and $\...
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Intuition: Dual Space is always Banach

Theorem. Let $X$ be a normed space and $Y$ be a Banach space. Then the set of continuous linear maps $L(X,Y)$ is a Banach space (with the operator norm). From the well-known theorem above, we get an ...
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Extreme points of $B_{C(K)^{*}}$

I am currently working on the following problem. Let $K$ be any compact Hausdorff space. Show that any extreme point of $B_{C(K)^*}$ is of the form $\pm \delta_s$ where $\delta_s$ is the probability ...
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1answer
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Show that collection of Lipschitz functions with Lipschitz norm is a Banach space

Question: Let $\mathcal{L}$ be the normed space of all Lipschitz functions on a Banach space $X$ that are equal to $0$ at the origin, under the norm $$\|f\| = \sup\left\{ \frac{|f(x)-f(y)|}{...
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Operator norm and Inequality Saturation

Let $T: X \to Y$ be a bounded linear map between Banach spaces $X,Y$. Generally, the operator norm is something that is notoriously difficult to compute, as we have two things to show. First we need: $...
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1answer
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Metric projection from space of bounded functions to finite-dimensional linear space

Apologies if the answer is obvious or should be easy to find, but so far I've had no luck. Let $X$ be a subspace of $\mathbb{R^k}$ for a finite $k$ and let $\mathcal{B}(X)$ be the Banach space of ...
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Is a Schauder Basis a minimal total set?

Given that the span of Schauder Basis - $S\subset V$ is $V$ itself, this implies that it is a total set. My teacher said that it was minimal as well but did not go on to prove it. I suspect it might ...
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Dunford-Pettis for Banach-Spaces

Can anyone tell me if the Dunford-Pettis property is met for a separate refelxive Banach space $X$ with dual $X'$? I would say that this is the case.
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Are there closed subspaces of the space of all bounded signed measure on $\mathbb{R}^d$ that are $\zeta$-biconvex?

Take $(\mathbb{R}^d , \mathfrak{B})$ as measurable space, and denote $\mathcal{M}(\mathbb{R}^d)$ be the linear space of all bounded signed measure. This is known to be a Banach space w.r.t. the total ...
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Sums of vectors in $l_p$

Suppose $x$ and $y$ are two vectors in $l_p$ (where $1\leq p<\infty$) such that $||x||=||y||=1$. Can we find a complex scalar $\alpha$, with $|\alpha|<1$ such that $||x+\alpha y||>1$. I don't ...
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Two versions for 1st Isomorphism Theorem for Banach Spaces

I have encountered two different versions for the First Isomorphism Theorem in the context of Normed/Banach spaces, I wanted to ask whether one implies the other. Theorem 1: Let $X,\,Y$ be normed ...
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example of a Banach space X and a subspace Y such that Y is strongly closed but not wealky closed.

I am trying to solve this exercise: find a Banach space X and a subspace Y such that Y is strongly closed but is not wealky closed. I know Y can't be a convex subspace because strongly closed + ...
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1answer
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Find the orthogonal complement on $L^2[0,1]$ of all polynomials.

The problem: Determine the orthogonal complement on $L^2[0,1]$ to all polynomials. My approach and intuition thus far: I know for sure intuitively that the orthogonal complement would just be the ...
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1answer
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Constructive proof of Banach-Alaouglu

Is there a constructive (i.e. not using Axiom of choice, and at most Axiom of dependent choice) proof of Banach-Alaoglu theorem in the case of separable Banach spaces. Even if it is needed assume that ...
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Banach spaces and isomorphism between dual spaces

Maybe is a silly question, but I have got a doubt about it: Let $(X,\Vert\cdot\Vert_X)$ and $(Y,\Vert\cdot\Vert_Y)$ be two linearly isomorphic Banach spaces. Then we know that their dual spaces are ...
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$\prod_i V_i$ is also a sub space of $B(H,K)$

Suppose $V_i$ are subspaces of $B(H_i,K_i)$. Is it true that $\prod_i V_i$ is also a sub space of $B(H,K)$ for some appropriate $H$ and $K$? I think we need to take $H$ and $K$ to be direct product ...
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1answer
27 views

reverse norm inequality when the Minkowski sum $Y+Z$ is closed

Problem Statement Let $(X,\|\cdot \|)$ be a Banach space and $Y$, $Z$ closed subspaces of $X$. If $Y+Z$ is closed, then show that there exists $C>0$ such that for all $x \in Y+Z$, there exist $y \...
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1answer
20 views

Inner product on a sequence and its limit

I am stuck on a question, and it seems like I'm missing a really obvious Cauchy-Schwarz application or something, but I am left scratching my head. Let $(x_n):n \in \mathbb{N}$ be a sequence in a ...
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1answer
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Representation of linear operator between $L^p$ spaces.

I was wondering where I could find a reference to the a characterization of continuous linear operators: $$T:L^p(X,\mu)\to L^q(Y,\eta)$$ of the form $T(f)(y)=\int_{X} k(x,y)f(x)d\mu$ for some $k$ ...
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1answer
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Is $C_0(\mathbb{R}^n)$ a Banach Space?

By defining $$ C_0(\mathbb{R}^n):=\{u:u\in C(\mathbb{R}^n),\quad\mathtt{and}\quad\lim_{|x|\rightarrow\infty}u(x)=0\} $$ normed with $||u||:=\sup_{x\in\mathbb{R}^n}|u(x)|$. As far as I can remember, ...
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1answer
25 views

Closure of $C^1[0,1]$ functions under the Lipschitz norm

I have been trying to prove that $C^1[0,1]$, i.e the space of continuously differentiable functions is closed under the $C^{0,1}[0,1]$ norm. $$||f||_{C^{0,1}}=||f||_{C^0}+\sup_{x\ne y}\frac{|f(x)-f(y)|...
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Closure of finite span has non-empty interior

Be $X$ a Banach space with countable basis. Suppose $(x_n)_{n \in \mathbb{N}} $ is a sequence in $X$ (allowing repetitions), such that $$(\forall x \in X)(\exists M_x \subseteq \mathbb{N})\left(|M_x|...
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1answer
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A sequence of bounded $C^1$ functions whose derivatives are unbounded.

What is an example of a sequence of functions, $(f_n)_{n=1}^\infty\subset C^1([a,b])$, which are bounded in $C^1([a,b])$ under $\|\cdot\|_{\infty}$ but are such that their first derivatives $\|f'_n\|_{...
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1answer
21 views

Regarding equivalent conditions of Frechet differentiability

Gateaux and Frechet differentiability in a Banach space are defined as below. Can you tell below how (ii) implies (i). The rest is easy.
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20 views

Dualilty map in Banach space

Let $X$ be a Banach space, we define $ \phi_-(y)=\lim_{t\rightarrow{0^+}}=\frac{|x|-|x-ty|}{t} $ $\phi_+(y)=\lim_{t\rightarrow{0^+}}=\frac{|x+ty|-|x|}{t} $ Then $ M^*(x)= \{ x* \in X^*: \phi_-(y)\...
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25 views

Weak convergence and convergence in norm implies convergences in a locally uniformly convex Banach space. [duplicate]

Let $X$ be a locally uniformly convex Banach space, then if $x_n \overset{w}{\rightarrow}x $ and $|x_n| \rightarrow |x| $ implies $x_n \rightarrow x$. $X$ is locally uniformly convex, Does the same ...
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2answers
62 views

$\Bbb R^d$ with norm is a Banach

Another problem I found in a textbook when brushing up on my real analysis. Was given a hint too. Prove that $\Bbb R^d$ with the norm $||x||_1=\sum_{i=1}^d |x_i|, \ x ∈ \Bbb R^d$ is a ...