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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What are the limitations of polyhedral norm?
Polyhedral norm is ||(x_1,x_2,...,x_n)||(k) =max(1≼i_1<...<i_k≼n)__[ε_i∈{-1,1}] (ε_1 x_(i_1)+ε_2 x_(i_2)+...+ε_k x_(i_k)
[I am giving the latex code for better understanding of the norm |(x_1, ...
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Existence and continuity of inverse operator
Let $E$ and $F$ be normed spaces. Let $T:E \to F$ is a linear operator and suppose that exists $c>0$ such that
$$\|T(x)\| \geq c \|x\|, \quad \forall x \in E.$$
Then it's easy to see that $\...
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Why metric equivalence does not preserve completeness but norm equivalence does? [duplicate]
I was studying Functional analysis when I came across the normed space equivalence i.e. equivalent norms. I already proved that equivalent norms on a vector space $X$ over field $\mathbb{R}$ or $\...
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Bounds on the norm of Lie Bracket with respect to Killing Form
If we have a Lie algebra $(V, [,])$ then the Lie bracket is a bilinear map
$$
[, ]: V \times V \to V
$$
Assuming $V$ is finite dimensional, this map is bounded and so, for any choice of norm, there ...
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Complete space relative to specific norm
Let $X=\{f\in C^1([0,\infty, \mathbb{R})): \lim_{t\to\infty}\frac{f(t)}{1+t}=\lim_{t\to\infty} f'(t)=0\}$. How to prove that $X$ is Banach space, if norm is defined as
$\|f\|=max(\sup_{t\geq 0}\frac{|...
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Prove: Bounded Linear transformation T hold norm for operator $\sqrt{(Tf_1)+(Tf_2)+...+(Tf_N)}$ [closed]
Suppose T is a bounded linear transformation mapping the space of real-valued
$L_p$ functions into itself with $||T(f)(x)||_{L_p} \leq M||f(x)||_{L_p}$.Prove:
Let $T^{'}$ be the extension of $T$ to ...
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Ratio of l1/l2 norm over n dimensional vector
According to here
$$\frac{(|c_1|+|c_2|+\cdots+|c_n|)^2}{c_1^2+c_2^2+\cdots+c_n^2}\geq 1$$
The equality holds when for each $i\neq j\in[n]$, $|c_i||c_j|=0$.
Could we improve the lower bound by ...
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If two norms induce the same topology on a vector space $V$ over a field $K$ with an absolute value, then they are equivalent
We say two norms $\|\cdot\|_1,\|\cdot\|_2$ are equivalent if there exists reals $c,C>0$ s.t. $c\|x\|_2 \leq\|x\|_1 \leq C\|x\|_2$ for all $x\in V$.
If two norms $\|\cdot\|_1,\|\cdot\|_2$ induce the ...
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Are these two Hölder-type inequalities valid?
Hereinafter, we use $p$-norm ($p\geqslant1$) in $n$-dimensional Euclidean space, which is given as $\|x\|_p=(\sum_{i=1}^n|x_i|^p)^{1/p}$.
As we know that, the following inequality holds for all $x\in\...
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Boundedness of the Laplacian Eigenfunctions
I have a doubt regarding the Laplacian eigenfunctions $\left\{\phi_n\right\}_{n=1}^\infty$ with Dirichlet boundary conditions. I know that the functions form an orthonormal basis in $L^2$ and an ...
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Prove that $\{f_n : n \in \mathbb{N}\}$ is closed in $C([0,1], \mathbb{R})$
Let $X = C([0,1], \mathbb{R})$ with the supremum norm: $||f|| = \sup\{\lvert f(x)\rvert : x \in [0,1]\}$. Define the functions $f_n \in X$ by
$$
f_n(x) =
\begin{cases}
nx & \text{if } 0 \leq x ...
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Characterizing Strict Convexity in $B^* $-Spaces over Complex Fields
Statement:
Let $ X $ be a $B^*$-space over the complex field $ \mathbb{C} $. We aim to prove that $ X $ is strictly convex(i.e.for any $ x, y \in X $ where $ x \neq y $ and $ \|x\|, \|y\| = 1 $, ...
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Proving a sequence of functions is Cauchy, but not convergent in a normed space.
$$f_n : [0,2] \to \Bbb R : x \mapsto f_n(x)= \begin{cases} x^n & \text{if } x \in[0,1] \\ 1 & \text{if } x \in (1,2] \end{cases}$$ for $$n \in \Bbb N$$. Show that the sequence $$f_n$$ is a ...
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Continuity on countably-normed Hilbert spaces
i was studying some Quantum Mechanics from this doctorate's work http://galaxy.cs.lamar.edu/~rafaelm/webdis.pdf and ata certain point, in Proposition 2 pag. 166 he means to prove the continuity of an ...
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Question about a set in a normed linear space
Let's say X is a normed, linear vector space over $\mathbb{C}$ that is complete with respect to the metric given by the norm. Let F be a subset of X that is closed and bounded. Question: is the set $\...
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$X+Y$ is closed iff $\inf\{||x-y||: ||x||=1, y \in Y\} > 0$
Prove that $X+Y$ is closed, where $X$ and $Y$ are closed normed subspaces of $Z$ and $X \cap Y = \{0\},$ iff $\inf\{||x-y||: ||x||=1, y \in Y\} > 0.$
This question looks easy but has me stumped ...
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For every functional on $l_p$ it exists a sequence that determines it uniquely
I am trying to prove this excercise from my Functional analysis book:
Let $p \in (1, + \infty ) \subset \Bbb{R}$, $q = (1- \frac{1}{p})^{-1}$ and denote $l_p$ the space of $p-$bounded sequences of a ...
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Seminorm of sum of two elements
I have a fundamental question about semi-normed spaces:
Let $(V, |\cdot|)$ be a semi-normed space and let $x,y \in V$ such that $|x| > 0 $ and $|y| = 0$. Can something be said about the seminorm of ...
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Completeness of finite dimensional normed vector spaces without using equivalence of norms
I believe completeness of finite dimensional normed vector spaces can be proven by showing completeness of all finite dimensional vector spaces under some conveniently chosen norm and then appealing ...
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Strict convexity of norm
Let $E$ be a Banach space and $B(E)$ (resp.~$S(E)$) be the closed unit ball (resp.~the unit sphere) of the Banach space $E$. $E$ has strictly convex norm if for each pair of elements $x, y \in S(E)$ ...
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Prove that $\langle x, y \rangle = \overline{\langle y, x \rangle}$
Let $X$ be a normed linear space over the field $\mathbb C$ with the norm $\|\cdot \|$. Let $x,y \in X$. Define $\displaystyle \langle x, y \rangle =\frac{1}{4} \sum_{k =0}^{3} i^{k} \Vert x +i^k y\...
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$S \in \mathcal{L}(L^1(\Omega))$, find $T^* \in \mathcal{L}(L^\infty(\Omega))$ with $T^*g = Sg \forall g \in L^1(\Omega) \cap L^\infty(\Omega)$
Below I will bring a passage from Heat Kernels by Wolfgang Arendt (Theorem 4.3.3, page 52). I need to understand it and write a more verbose report based on the chapter, however I am stuck at this ...
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dual space of l2 with strange norm
Consider $\displaystyle(\ell_2, \lVert\cdot\rVert_\star), \lVert x\rVert_\star = \sum\limits_{k=1}^{\infty}\frac{|x(k)|}{k}$. What is its dual space? Is this space reflexive?
My idea is to consider $\...
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Generalised directions
Let $X$ be a normed space, $A$ a real vectorspace. Let a map $\phi:X \setminus \{0\} \to A$ such that it satisfies the following properties
$\phi(\lambda x) = \phi(x)$ for every $\lambda >0$
if $\|...
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let X is a normed vector space. if $ D \subseteq X$ is a balanced set $ D^0 \cup \{0\}$is a balanced set.
I tried to prove that:
"Let $X$ be a normed vector space. If $D \subseteq X$ is a balanced set then $D^0 \cup \{0\}$ is a balanced set."
$D^0$ is the interior of $D$.
I tried to prove it ...
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"Best" Submultiplicative / Subordinate norm?
I have $y = Ax$ where x, y are vectors and $A$ is a matrix.
I want to get the best $K$ such that $||y|| \leq K||x||$. Ideally, $K$ is a matrix norm. Especially, $K$ can be a subordinate matrix norm. I ...
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Papa Rudin $6.16$ theorem case 2.
There is the theorem:
Suppose $1\leq p \lt \infty $, $\mu$ is a $\sigma$-finite positive measure on $X$, and $\varphi$ is a bounded linear functional on $L^{p}(\mu)$. Then there is a unique $g \in L^{...
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Product of commutative symmetric positive definite matrices
Let A and B be commutative symmetric positive definite matrices. The goal of the question was to show that $||x||_{A,B}=\sqrt{<Ax,Bx>}$ is a norm in $\mathbf{R}^n$, where $<x,y>$ is the ...
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The closed unit ball of a normed space X is compact iff X is finite-dimensional
I have a question about this theorem:
" The closed unit ball of a normed space X is compact iff X is finite-dimensional."
if I take (R,+,.,Q)
R is a infinite dimensional vector space when ...
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$f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$ $\Rightarrow$ $\nabla(f_kg_k)\rightharpoonup \nabla(fg)$ in $L^p(\Omega)$
Suppose that $\Omega \subset \mathbb{R}^d$ is bounded with a Lipschitz boundary and $f_j \rightharpoonup f$ and $g_j \rightharpoonup g$ weakly in $H^1(\Omega)$. Show that, for a subsequence, $\nabla(...
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Papa Rudin $6.16$ theorem.
There is the theorem:
Suppose $1\leq p \lt \infty $, $\mu$ is a $\sigma$-finite positive measure on $X$, and $\phi$ is a bounded linear functional on $L^{p}(\mu)$. Then there is a unique $g \in L^{q}(\...
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Inequality regarding Matrix Norm and Inverse Matrix
Currently, I'm stuck to one of a statement in a paper. Following is a brief summary of the paper regarding my question. (although the topic of the paper is mainly statistics, the question purely ...
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Inequality for inverse of an unbounded self-adjoint operator
Given an unbounded self-adjoint operator $A$ on some Hilbert space $\mathcal{H}$, and $\mu$ a non zero real number: I want to show that
\begin{equation}
\lVert (\mathbf{A} + i\mu \mathbf{I})^{-1} \...
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2
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Derivative in normed space
I'm studying the Differential Calculus Functions of several variables. Let $f:A\subset\mathbb{R}\to\mathbb{R}^n$ is differentiable with $||f(t)||>0$ for all $t\in A$. Prove that: $$u(t)=\dfrac{f(t)}...
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How do you present the absolute value as a norm and derive results?
I asked myself a question while reading the post: Absolute value of complex numbers $|a+bi|$ .
Let us take Euclidean space $\mathbb R^2$, with dot product noted $\langle .,.\rangle:\mathbb R^2 \times \...
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Matrix norm that keeps order relation when applied to a vector
Let $M_{n\times n}$ be the vector space of real square matrices of size $n\times n$. Does there exist a matrix norm $\left\lVert\cdot\right\rVert_M$ such that if $\left\lVert A\right\rVert_M \leq \...
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Jacobian of seminorm on $\mathbb R^n$
As follows $\mathcal{H}^n$ denotes the $n-$dimensional Hausdorff measure and $\omega_n$ denotes the volume of the unit ball on $\mathbb R^n$.
In the article Rectiable sets in metric and
Banach spaces ...
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Necessary and sufficient conditions for a product of norms to be the square of a norm
Let $(E,N)$ be a normed space, and $M$ be another norm. Can you find a necessary and sufficient condition on $M$ so that $\sqrt{NM}$ is also a norm on $E$?
I hink the condition is that $M$ is ...
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Help Needed: Proving Equivalence of Inner Product and Parallelogram Law [duplicate]
Problem: Let $(X, || \cdot ||)$ be a normed space. Prove that the following are equivalent:
i) There exists an inner product $< \cdot , \cdot >$ on $X$ such that $||x|| = \sqrt{< x, x >}$ ...
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(Why) is the norm of a RKHS positive definite?
$
\newcommand{\real}{\mathbb{R}}
$
A $\color{red}{\text{(strictly)}}$ positive definite kernel $k: \real^d\times \real^d \to \real$ satisfies for all $x_i \in \real^d$, $a=(a_1,\dots, a_n)\in \real^d$
...
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Show that $f$ is differentiable at $a$ if and only if $f(tx+(1-t)a)$ has derivative at $0$ for all $x$
In the book "A Graduate Course on Statistical Inference" (as you can see in this link) there's the following passage: "[a function] $f(\theta )$ is differentiable at $\theta _0$ if and ...
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Uniqueness of extension of the normed space structure to a larger topological space
The following popped up when I was contemplating the completion of normed linear spaces.
Question: Let $E$ be an NLS (normed linear space) and $X$ a topological space. Let $E$ sit inside $X$ as a ...
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Determining the norm of a linear operator of a normed R-vector space.
I have been trying to solve an exercise in Normed Vector Spaces, and I'm stuck in the 2nd question.
My answer to the 1st question:
We have $\varphi$ linear.
Let
$$||\varphi(f)||_{1} = \int_0^1 |\...
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answer
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Proving a characterization of the Dense Linear Subspaces on a Normed Space
Let $(E, \lVert \cdot \rVert)$ a normed space over a field $\Bbb{K}$ and $M$ a linear subspace of $E$ . Then, $$\overline{M}=E \iff \forall f \in E' : f(M) = \{0\} \rightarrow f(E) = \{0\}$$
($E'$ is ...
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Sub-multiplicative property of matrix norms
I just wanted to clarify my understanding of the sub-multiplicative property of the matrix operator norm.
On this Wikipedia page it says for any matrix:
$
\|AB\|_{\alpha, \gamma} \leq \|A\|_{\beta, \...
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On a property for normed spaces
I came upon the following specific property for a normed space $X$, and I am looking for a characterization of the normed spaces where it holds true:
If a sequence $x_n$ in $X$ satisfies $\...
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unbounded operator satisfying $||T(x_n)|| \to \infty$
Let $E,F$ be a normed space, and
$T:D(T) (\subset E) \rightarrow F$
be a densely defined unbounded linear operator.
By unbounedness,
for all $x \in E$, there is a sequence $(x_n) \subset D(T)$ such ...
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Equality in a norm in $\mathbb{R}^n$ that is not strictly convex
If a norm $\lVert \cdot \rVert$ is strictly convex and $\lambda \in (0,1)$, then
$$
\lVert \lambda v + (1-\lambda)w \rVert = \lambda \lVert v \rVert + (1-\lambda)\lVert w \rVert \tag{1}\label{1}
$$
...
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1
answer
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Any norm on $\mathbb R^n$ induces product metric?
Let $X_1, \ldots, X_n$ be metric spaces and $\|\cdot\|$ be a norm on $\mathbb R^n$. Then it's easily seen that if $\|\cdot\|$ is monotonic in each coordinate while (others fixed) in the orthant $[0, +\...
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To verify whether Banach space
($X,||.||$) is a Banach space and $X=M*N$, where M and N are closed subspaces of X. Let $x \in X$ have a unique representation $x=m+n$, $m \in M$ and $n \in N$
Define $||.||_{1}$ : $X \to R$ by $||x||...
