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Questions tagged [normed-spaces]

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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Given that $x_i\ge0$ and $0<p<1$, find an upper bound for $\sum_{i=1}^n x_i^p$ as a function of $a:=\sum x_i$ [duplicate]

Suppose $\boldsymbol{x}$ is an $n$-dimensional vector and all elements are nonnegative with the condition that $\sum_{i=1}^nx_i= a$. I am wondering how it is possible to find an upper bound on $\sum_{...
Amin's user avatar
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There is at least one point of every non-empty open subset of the $\ell^2$ space whose first coordinate is nonzero

Here we take $$ \mathbb{N} := \{ 1, 2, 3, \ldots \}. $$ Let $\ell^2$ denote the set of all the real (or complex) sequences $\left( \xi_i \right)_{i \in \mathbb{N} }$ such that the series $\sum \left\...
Saaqib Mahmood's user avatar
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Multiplicity of bilateral shift on a Banach space

Let $\mathbb{X}$ be a Banach space. A bijective linear map $V: \mathbb{X} \to \mathbb{X}$ is said to be a bilateral shift if there is a closed subspace $\mathbb{L}$ of $\mathbb{X}$ such that $\mathbb{...
swapan Jana's user avatar
2 votes
2 answers
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Bounding $\Vert f\Vert \Vert g\Vert$ by $\Vert wf \Vert^2 +\Vert w^{-1} g\Vert$

Let $\Omega=[0,1]^d$ for some $d\ge 1$, and let $w:\Omega \to (0,\infty)$ be a continuous function. Is is true that $$\Vert w f \Vert_{L^2(\Omega)}^2+ \left\Vert \frac{1}{w}g \right\Vert_{L^2(\Omega)}^...
Tulip's user avatar
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Convex combination of equidistant curves

Say we have three curves $\gamma, \delta, \varepsilon : \mathbb R \to \mathbb R^n$ such that the distances $\lVert \gamma(t) - \delta(t) \rVert$ and $\lVert \gamma(t) - \varepsilon(t) \rVert$ are ...
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Laplacian of arbitrary power of arbitrary norm

So I have the function $f : \mathbb R^d \to \mathbb R$ given by $f(x) = \lVert x \rVert_p^q$, where $\lVert \cdot \rVert_p$ denotes the $p$-norm on $\mathbb R^d$, given by $$ \lVert x \rVert_p = \left(...
markusas's user avatar
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Which metrics (on vector spaces) can be induced?

Is there a way to classify which metrics defined on vector spaces can be induced by a norm? ie. there exists norm $n: X\to \mathbb{R}$ on the vector space $X$ such that the metric $d(x,y)=n(x-y)$. I ...
HIH's user avatar
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Isomorphism between linear functionals $\mathbb{L}(\mathbb{R}^2)$ and $\mathbb{R}^2$.

Problem: Show that if $\mathbb{L}(\mathbb{R}^2)$, the space of the linear functionals in $\mathbb{R}^2$, with the matrix norm $\|\cdot\|_p$ ($p > 1$) is isomorphic with the $\mathbb{R}^2$ space ...
Wellington Silva's user avatar
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Prove that $BE^{\alpha}$ is a Banach Space

Let $E$ a Banach space, can we prove that the set $$BE^{\alpha}=\{f \in S': \ \sup_{t>0}t^{\alpha}\|G(t)f\|_{E}\}$$ is also a Banach space? Here $S'$ is the set of tempered distributions $S'(\...
Jarbas Dantas Silva's user avatar
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If a pseudonorm function $N$ is continuous in a given topology, does the pseudometric topology formed by $d(x,y)=N(x-y)$ coincide with first topology?

Define $p_n\#= p_n p_{n-1} \cdots p_1$ for $n = 0$ to be $1$, then we have a function: $$ N : \Bbb{Z} \to \Bbb{Z}, \\ N(x) = \left |-1/2 + \sum_{d\ \mid\ p_n\#} (-1)^{\omega d} \sum_{r^2 = 1 \mod d} \...
SeekingAMathGeekGirlfriend's user avatar
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Triangle inequality for $l^2$ norm

While working trough exercises in Peter Lax' book on Functional Analysis, I got stuck on the following exercise from section 5: Exercise 1: Show that $|(z,u)|'=(|z|^2+|u|^2)^{1/2}$ is a norm on $Z\...
JackpotWizard 180's user avatar
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$||Ax|| \leq ||x||_{*} \leq \sqrt{n} ||Ax||_2$

Show that exists a norm $||.||_{*}$ such that for each matrix satisfying $$||Ax||_2 \leq ||x||_{*} \leq \sqrt{n} ||Ax||_2$$ exists a vector $x$ such that $||x||_{*} = \sqrt{n}||Ax||_2$. This is ...
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The volume of the unit ball in $L((\mathbb{R}^n, \| \cdot \|_2), (\mathbb{R}^m, \| \cdot \|_2))$ with the operator norm

Consider the normed space $(X_{n, m}, \| \cdot \|) = L((\mathbb{R}^n, \| \cdot \|_2), (\mathbb{R}^m, \| \cdot \|_2))$ of linear operators $\mathbb{R}^n \to \mathbb{R}^m$ endowed with the operator norm....
Smiley1000's user avatar
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Prove that $A$ is a linear operator mapping from $\ell^2$ to $\ell^2$. Determine $\|A\|$, $A^*$ and $\sigma(A)$.

For every $x = (x_n) \in \ell^2$, let $$ Ax = \left(x_1, \frac{x_2}{2}, x_3, \frac{x_4}{2^2}, x_5, \frac{x_6}{2^3}, \ldots \right). $$ Prove that $A$ is a linear operator mapping from $\ell^2$ to $\...
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Linear Analysis – Examples 1-Q5 General $l^p$ spaces vs $l^1, l^\infty$

Let $1<p<\infty$, and let $x$ and $y$ be vectors in $l_p$ with $\left \|x \right \|=\left \|y \right \|=1$ and $\left \|x +y \right \|=2$, how to prove $x=y$? I know how to prove for $p=2$ ...
HIH's user avatar
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For a given $k \in \mathbb{N}$, define the mapping $A: X \to X$ by the rule $ (Af)(x) = x^k f(x). $

Let $X = (C[-1,1], \|\cdot\|_\infty)$. For a given $k \in \mathbb{N}$, define the mapping $A: X \to X$ by the rule $ (Af)(x) = x^k f(x). $ (a) Prove that $A$ is a bounded linear operator. (b) ...
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If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space.

I am wondering if the following statement might hold (as I wanted to use this in solving another problem): If $ \phi: X \to X^* $ is an isometry, then $ X $ is a complete space. I know that $ X^* $ is ...
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I want to prove that $T$ belongs to $B(X)$, meaning it is a bounded linear operator.

Let $T_f: X \rightarrow X$ be defined by $T_f(x) = f(x) u$ for every $f \in X^*$ for some non-zero $u \in X$. I want to prove that $T$ belongs to $B(X)$, meaning it is a bounded linear operator. I ...
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3 votes
2 answers
113 views

Nature of the Euclidean Norm

I've been re-reading my linear algebra book and a definition is given of the norm of a vector in $\mathbb{R}^n$ to be: ||v|| $= \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$. For $\mathbb{R}^2$ and $\mathbb{...
MattKuehr's user avatar
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Showing $\|Ax\|_2\leq\|x\|_*\leq\sqrt{n}\|Ax\|_2$

Show that for any real norm $||x||_{*}$, exists a matrix such that it holds $$||Ax||_{2} \leq ||x||_{*} \leq \sqrt{n} ||Ax||_{2},$$ finding an example where the upper bound it’s tight. I feel this ...
jacopoburelli's user avatar
2 votes
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50 views

Problem in A Course in Approximation Theory by Cheney and Light

In chapter 2 of "A Course in Approximation Theory" by Cheney and Light, problem 11 states: How large can the coefficients be in a polynomial $p$ of degree at most $n$, if $p$ satisfies the ...
tox123's user avatar
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Conceptual difference between regularized and constrained optimization.

In the literature of compressed sensing, one encounters optimization problems as shown in equation (1) and (2). focusing on the conceptual differences between regularized optimization and constrained ...
ACR's user avatar
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Prove that for every $n \in \mathbb{N}$, the mapping $A_n$ is a bounded linear operator from $C[0,1]$ to $C[0,1]$ and calculate its norm.

Let $C[0,1]$ be a normed space equipped with the norm $\|\cdot\|_\infty$, and let for every $n \in \mathbb{N}$, the mapping $A_n$ be given by the prescription $ (A_n(f))(x) = \begin{cases} f(x), &...
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Example of Hilbertian norm on the space of radon measures

Assume we are given compact subset $X \subset \mathbb{R}^n$. Consider the space of radon measures $M(X,\mathbb{R})$. I'm trying to find some Hilbert norm on this space either globally or locally. I ...
supernova's user avatar
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Upper bound on infinity norm from Sobolev W^1,2 norm in >1 D? [duplicate]

Let $\Omega:=[0,1]^d, d\in\mathbb{N}$. Let $u\in W_0^{1,2}(\Omega)$, the 0 subscript meaning $u$ is 0 on $\partial\Omega$. Assume also $$ \int_{\boldsymbol x\in\Omega}\nabla u(\boldsymbol x)\cdot\...
Leon Avery's user avatar
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Norm on the space of vector fields

Let $M$ be a finite dimensional manifold. There exists the notion of the norm of a tangent vector field on M? In other words, the space of tangent vector fields is a normed space? Thank you in advance ...
V.Jiménez's user avatar
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1 answer
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Prove that $A$ is a bounded automorphism of the vector space $X$. Also prove that $\sigma(A) = \{ \lambda \in \mathbb{C} \mid |\lambda| = 1 \}$.

Let $X = \{ f : \mathbb{R} \to \mathbb{C} \mid f \text{ is a bounded function} \}$. We know that $X$ is a Banach space equipped with the norm $\| f \| = \sup_{x \in \mathbb{R}} | f(x) |$. Let $c \in \...
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1 answer
41 views

Prove that the sequence $(A_n(f))$ is convergent in the normed space $C[0,1]$. Prove that the sequence $(A_n)$ is not convergent in the operator norm.

Let $C[0,1]$ be a normed space equipped with the norm $\|\cdot\|_\infty$, and let for every $n \in \mathbb{N}$, the mapping $A_n$ be given by the prescription $ (A_n(f))(x) = \begin{cases} f(x), &...
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1 vote
0 answers
55 views

Spanning set of support functionals in dual space

I am currently studying about supporting hyperplane (or, support functional) in dual space. Since, I am new in these topics I met with the following queries: Let $X$ be a normed space and $X^*$ be the ...
Tutun's user avatar
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3 votes
1 answer
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The space $C_c$ of real-valued compactly supported, continuous functions is not a Banach space under any norm

This answer showed the space $c_{00}$ of compactly supported sequences, is not a Banach space under any norm. I wonder if the same is true for the space $C_c$ of real-valued compactly supported, ...
hbghlyj's user avatar
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1 answer
70 views

(Copy) Set of linear functionals span the dual space iff intersection of their kernels is {0} .

I have fully understood the following question and got a motivation from it. Set of linear functionals span the dual space iff intersection of their kernels is $\{0\}$. My question is what will be the ...
Tutun's user avatar
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-1 votes
2 answers
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Why is positive definite defined this way?

Norm: A norm on a vector space $V$ is a function $\| \cdot \| : V \to \mathbb R$ which assigns each vector $x$ its length $\|x\| \in \mathbb R,$ such that for all $\lambda \in R$ and $x, y \in V$ the ...
nameless___'s user avatar
3 votes
1 answer
109 views

How to show that a subset of a normed space isn't dense?

Consider an arbitrary normed vector space $(X, \Vert \cdot \Vert_X)$ and let $Y \subset X$. We say that $Y$ is dense in $X$ if and only if $$ \forall x \in X, \epsilon > 0, \, \exists y \in Y : \...
xyz's user avatar
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find $ker f$ for this function

We Define the normed vector space of bounded sequences in $ \mathbb C$ as follow: $$ l_\infty := \{ x={x_k} \subset \mathbb C \mid \sup \{ |x_k| : k \in \mathbb N \} < \infty \}$$ $$ \|x\|=\sup \{ ...
A12345's user avatar
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1 answer
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Let $X$ be a reflexive space and let $f \in X^*$, $ Px = x - \frac{f(x)}{\|f\|} y, \quad x \in X. $

Let $X$ be a reflexive space and let $f \in X^*$. (a) Show that there exists $y \in X$ such that $x - \frac{f(x)}{\|f\|} y \in \ker f$ for every $x \in X$. (b) Let $P : X \to X$ be the mapping defined ...
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1 vote
1 answer
72 views

Let $(X, \|\cdot\|)$ be a normed space, $a_1,a_2,\ldots, a_n \in \mathbb{C}$ and $x_1,x_2,\ldots, x_n$ linearly independent vectors of the space $X$.

Let $(X, \|\cdot\|)$ be a normed space, $a_1, a_2, \ldots, a_n \in \mathbb{C}$ and $x_1, x_2, \ldots, x_n$ linearly independent vectors of the space $X$. Show that the following two statements are ...
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1 answer
33 views

$\|f^{-1} \|= \frac{1}{a} $

We define $$B(X,Y)= \{ f \mid f:X \to Y, \text{ is continuous and linear function}\}$$ Let $X,Y$ bee Banach normed spaces, $f \in B(X,Y)$, $a>0$, and $ \|f(x)\| \ge a\|x\|$ for all $x\in X$. Then $...
A12345's user avatar
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For normed spaces $ E=\{x\in X \mid \inf\{ \|x-e\| \mid e \in E \}=0 \} $

Let X be a normed vector space and $ E \subset X$ Prove that $$ E=\{x\in X \mid \inf\{ \|x-e\| \mid e \in E \}=0 \} $$ I tried to prove like this: Let $\begin{align*} x \in E &\Rightarrow 0 \le \...
A12345's user avatar
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1 answer
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We are interested in finding for which $\lambda$ the operator $A - \lambda I$ is not surjective.

We are in the space $X = C[1/2, b]$ for some $b < 1$. We are interested in finding for which $\lambda$ the operator $A - \lambda I$ is not surjective. The operator $A: X \to X$ is given as $(Af)(x) ...
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-2 votes
1 answer
117 views

Thus $MV - VM = V^2$. So the spectrum of $V^2$ is $\sigma(V^2) = (\sigma(V))^2=0$. Why??

I am curious if this statement holds (it doesn't make much sense to me, but it was written in solutions in this form): $\sigma(A)=0\implies\sigma(A^2)=(\sigma(A))^2=0.$ Can anybody explain to me why ...
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1 answer
34 views

About strictly convex norm

Let X be a normed vector space. $\| . \|$ is a norm. we said this norm is a strictly convex norm if $$ \forall x,y \in X : \| x\| \le 1, \|y\| \le 1 \Rightarrow \| \frac{x+y}{2} \| <1 $$ I have ...
A12345's user avatar
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1 vote
1 answer
30 views

$\|x\|_0 = \sup_{T \in G} \|Tx\|$ is equivalent to the norm $\|\cdot\|$ and in which all operators $T \in G$ are isometries.

Problem: Let $(X, \|\cdot\|)$ be a Banach space and let $G$ be a subset of $\mathcal{B}(X)$. Assume that $G$ is a group (under the operation of operator multiplication) such that $\|T\| \leq 1$ for ...
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2 answers
81 views

I want to prove the following: If $(X, \|\cdot\|)$ is a Banach space, then every Cauchy sequence in $(X, \|\cdot\|_1)$ converges in $(X, \|\cdot\|)$

Let $(X, \|\cdot\|)$ be a normed space, and let $A: X \to X$ be a linear operator. First, I proved that the prescription $\|.\|_1 = \|x - Ax\| + \|Ax\|$ defines a norm on $X$. I want to prove the ...
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0 votes
2 answers
76 views

Let $X$ be the vector space of all real sequences that have at most finitely many non-zero terms. Is $(X, \| \cdot \|)$ a Banach space?

Let $X$ be the vector space of all real sequences that have at most finitely many non-zero terms. I was able to show that the prescription $\| \{x_n\}_{n \in \mathbb{N}} \| = \max_{n \in \mathbb{N}} |...
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1 answer
31 views

linear vector addition in norm of vectors suggest a line segment in the unit circle

Let $V$ be a normed vector space, which contains two linearlly-independant vectors $x,y$ such that $\|x+y\|=\|x\|+\|y\|$. Prove that the unit circle $\{x \in V:\|x\|=1\}$ contains a line segment. From ...
Yotam Ohad's user avatar
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19 views

Bound on spherical harmonic L2 norm from L_inf norm

The online notes here state the following as proposition 6.0.1. If $f \in H_d$ then $$\sup_{x \in S^n} |f(x)| \leq \sqrt{\frac{dim H_d}{ \text{vol}(S^{n-1})}} |f|_{L^2},$$ where $H_d$ is the space of ...
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Reference for isomorphism theorem for Banach spaces

Let $X,Y$ be Banach spaces, and a linear continuous operator $T:X \rightarrow Y$ then we have that, denote by $N =\text{ker }T$ and by $\text{Ran }T$ the closed range of T then, $$ X / N \cong \text{...
Scottish Questions's user avatar
3 votes
1 answer
95 views

rational complex minmax

As the title says, I wanted to understand the following part of a proof in Guttel (Corollary 2, page 6). $f$ is an analytic function and $r_m \in \mathcal{P}_{m-1}/q_{m-1}$ is a rational function with ...
jacopoburelli's user avatar
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27 views

Does this relation between Wasserstein distances hold: $W_1(\mu,\nu)\leq W_2(\mu,\nu) \leq ... \leq W_\infty(\mu,\nu)$?

I stumbled upon this interesting statement in this paper: "One interesting observation is that the Wasserstein ambiguity set with the Wasserstein order p = 2 is less conservative, because the 2-...
osi41's user avatar
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-1 votes
1 answer
47 views

Is there a notation for normed spaces $(X, \| \cdot \|_X)$ and $(Y, \| \cdot \|_Y )$ such that $X = Y$ and $\| x \|_X = \| x \|_Y$ for all $x \in X$?

Suppose that we are dealing with two abstract nomed spaces $(X, \| \cdot \|_X )$ and $(Y, \| \cdot \|_Y)$ such that $ X = Y $ (that is, every element of $X$ belongs to $Y$ and every element of $Y$ ...
xyz's user avatar
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