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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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22 views

Norm $\mathcal{C}^1$ multivariate functions

Given an open and bounded subset $\,\Omega\subset\mathbb{R}^n\,$ and the interval $\,[0,T]\,$ such that $\,T<\infty$, I'm trying to find a norm for $\,X=\mathcal{C}^1\left([0,T]\times\Omega\right)$ ...
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1answer
25 views

For A,B subsets of a Normed Vector Space, A closed, B Compact, Show A - B Is Closed [duplicate]

Statement of the problem: Let $E$ be a Normed Vector Space over the real numbers. Let $A, B$ be subsets of $E$ such that: $A$ and $B$ are non-empty, $A \cap B = \emptyset $. Assume $A$ is closed and ...
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18 views

The relationship among different types of fundamental spaces.

I'm just looking to make sure my understanding of certain fundamental spaces are correct. Denote the set of all vector spaces by $V$, the set of all metric spaces by $M$, the set of all normed spaces ...
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21 views

How to prove Neumann series doesnt converge when spectral radius > 1?

For an operator $T \in B(X)$, its spectral radius is $$r(T) = \lim_{n\rightarrow \infty} \|T^n\|^{1/n}$$ and the Neumann series is $$\sum_{n=0}^{\infty}T^n.$$ If $r(T)>1$, I can show that series ...
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1answer
32 views

A necessary and sufficient condition on a normed space $X$ for the strong operator topology and norm topology on $B(X)$ to coincide

Is there a simple criterion that determines whether the strong operator topology and norm topology on $B(X)$ coincide, when $X$ is a normed space? If $X$ is a Hilbert space, the necessary and ...
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1answer
47 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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40 views

Convexity properties of a cone

Let $M$ be a normed space and $A$ a subset (nonempty) of the unit sphere ($S_1$, which is the points of norm $1$). Define, for $\alpha >0$, a $A^{\alpha}=\{x\in S_1 | d\left( x,A\right) \leq\alpha\}...
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13 views

Are normed spaces equipped with the weak topology sequential [duplicate]

Let $X$ be a normed space equipped with the weak topology. Is $X$ a sequential space (using this definition)? That is can we test closedness in the weak topology using weakly convergent sequences? I ...
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98 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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Squared norm of sum larger than sum of squared norms [closed]

Let $G$ be an inner product space. Show that for any $g_1,...,g_n\in G$, there exist scalars $c_1,...,c_n$ s.t. $\forall i, |c_i|=1$ and $$\left\|\sum_{i=1}^n c_ig_i \right\|^2\geq \sum_{i=1}^n\|g_i\|^...
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31 views

Closure of c$_0$ in Uniform Topology

The Uniform Topology is generated by the metric $d$ on the set $X=l^\infty$ $d(x,y)= \sup \{ |x_n -y_n|: n\in \mathbb{N}\} $ We now prove $\bar{ c_0} \subset c_0 $, let $x \in \bar{c_0}$ we have ...
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29 views

Lp , norm infinity, integrable function [closed]

neither I could proof it and nor find an counterexample for the following problem
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13 views

Finite Rank Operator in Normed Space, not necessarily Hilbert neither Banach

Suppose that $E$ and $F$ are normed spaces and $T:E \rightarrow F$ is a bounded linear operator. I NEED TO SHOW WHAT FOLLOWS: If there are $n\in \mathbb{N}, f_{1}, ..., f_{n}\in E^{\ast}$ (dual of $...
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1answer
28 views

Proving that $l_\infty$ is complete

I'm learning about Hilbert spaces and operators theory, from some book. I came across the following question - And the books' answer: What I don't understand in the proof - Why can we understand ...
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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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Simplifying induced matrix norm expressions

Notation Absolute value of $c\in\mathbb{C} = \left|c\right|$. Entrywise absolute value of $A = \text{abs}(A)$. Complex conjugate transpose of $A = A^* = (\overline{A})^T$. $A$'s $i$th singular value $...
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2answers
32 views

The rotation matrix is non-expansive

Define for each $\alpha\in \mathbb{R},$ $$A(\alpha)=\begin{bmatrix}\cos\alpha& -\sin\alpha\\ \sin\alpha &\cos\alpha\end{bmatrix}.$$ Then $A$ is non-expansive if $$\|A(\alpha) -A(\beta)\|\leq \|...
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34 views

Are the spaces $T_pM$ and $\mathbb R^n$ homeomorphic?

Let $T_pM$ be the tangent space at a point $p$ in a n-dimensional smooth manifold $M$. In addition, if we assume $(M,g)$ as a smooth Riemannian manifold, then $T_pM$ is a n-dimensional real normed-...
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36 views

Why are reflexive spaces called like that?

A normed spaces $(X, \| \cdot \|)$ is called reflexive if the evaluation map $X \to X^{**}$ is an isomorphism. If $X$ is reflexive, it's not analytically distinguishable from it's bidual space $X^{**}$...
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47 views

The strong, weak and weak-star topologies coincide on finite dimensional spaces.

Let $E$ be a finite dimensional normed linear space. I have been able to show using set inclusion that $s=w=w^*$, where $s,\,w$ and $w^*$ represent strong, weak and weak-star topologies, repsectively. ...
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2answers
90 views

Differentiability, linear operators

Let $Y$ be a complete normed linear space, and let $M$ denote the space of bounded linear operators from $Y$ to itself. Let $L : M → M$ be the map defined by $L(A) := A^2$. I am supposed to show that ...
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29 views

Alternatives to Gram-Schmidt

So I'm curious if what I'm about to say is well-known and/or true, because I don't really have time to investigate right now. By inspecting the graphs of sets of orthonormal polynomials (like ...
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37 views

Proof of the Alaoglu Theorem

I was reading through the proof of the Alaoglu theorem which states Let $X$ be a normed space Then the unit ball in $X^*=B^*$ is compact with respect to the $weak^*$ topology. The proof goes as ...
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2answers
49 views

Norm Topologies on an Infinite Dimensional Vector Space

I know that all norms on a finite dimensional vector space induce the same topology. Moreover, I know that there are infinite dimensional vector spaces with norms that don't induce the same topology. ...
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1answer
23 views

X- normed space, Y - proper subspace. There exists such y, that ||x-y|| = dist(x,Y).

Let $Y$ be a finite-dimensional subspace of a normed space $X$. Show that for every $x \in X$ there exists $y \in Y$ such that $\left\Vert x-y \right\Vert = \operatorname{dist}(x,Y)$.
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15 views

Is the space of all infinitesimal numeric sequences complete?

Is the space $c_0$ of all infinitesimal numeric sequences $a = (a_1, a_2,...,a_n,...)$ complete? The norm is $||a|| = sup|a_i|$. As i couldn't prove that every Cauchy sequence has a limit in $c_0$ (...
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64 views

For a general norm on $\mathbb{R}^d$, how can the intersection of a ball's boundary and a cone be contained in a $(d-1)$-hyperplane?

Consider $\mathbb{R}^d$ under a general norm (not necessarily Euclidean). Consider two fixed concentric balls centered at the origin $B(0, R_1)$ and $B(0, R_1/2)$. Let $x$ be an arbitrary point with $|...
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2answers
50 views

Is the set of inner product spaces a subset of the set of metric spaces?

I found this picture when looking up topological spaces. Is this picture actually supposed to be interpreted as decreasing sets? That is, all inner product spaces are normed vector spaces, all metric ...
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220 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 ...
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1answer
14 views

The angle between the $x$-axis and the vector that reachs the $2$-norm of a matrix

Suppose we have a matrix $A\in \mathbb{R}^{2\times 2}$. We have the $2$-norm defined as $\left \|A\right \|_2=\max_{\left \|x\right \|_2=1}\left \|Ax\right \|_2$. We know that in order to find a unit ...
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1answer
46 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 derivative. 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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1answer
16 views

Finding norm of a bounded linear functional

Let $X=\{x\in C[0,1]: x(0)=0\}$ with sup norm and let $f:X\to \mathbb{K}$ be defined by $$f(x)=\int\limits_0^1 x(t)dt \text{ for all }x\in X.$$ I want to show that $\|f\|=1$. It is easy to show that $\...
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72 views

Is the space of real sequences normable

Intuition says the vector space of real sequences $R^N$ ($N$ the natural numbers, pointwise addition of real coordinates) is not normable. I have found this surprisingly hard to prove. I am aware ...
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36 views

Properties of convergence in $L^{\infty}$

Let $\Omega \subset \mathbb{R}$ be a bounded domain and $\alpha > 0$ be fixed. Assume that $|| u_{n} - v||_{L^{\infty}(\Omega)}\to 0$ as $n\to\infty$. How can I show that $||\, |u_{n}|^{\alpha} - |...
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Showing $f(B_r)$ is relatively compact for a certain $f$.

Let $X$ be a Banach space and $g:B_r\to X$ a continuous function, where $B_r:=\{x\in X\mid \|x\|\leq r\}$. Suppose that $g(x)\neq 0$, for all $x\in B_r$. $g(B_r)$ is relatively compact. Define $f:...
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1answer
33 views

False statement that all norms on the direct sum of normed spaces are equivalent.

I am currently working on the exercises in Conway's A Course in Functional Analysis and I think the following problem is not true. Here $\oplus_p X_k = \{(x_1, ..., x_n) \in \oplus_{k=1}^n X_k: (\...
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42 views

example of a sequence in $C[0,1]$ which converges in weighted sup norm but not in L1 norm

I am trying to think of a sequence $\lbrace f_n \rbrace$ in $C[0,1]$ which converges in the weighted supremum norm $$\|f\| = \sup_{x \in [0,1]}|xf(x)|$$ but not in the L1 norm $$\|f\|_1 = \int_0^1 |f(...
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36 views

The span of a finite number of vectors in a normed vector space is closed

I want to prove that the span $S$ of a finite number of vectors $v$ in an arbitrary normed vector space $V$ is a closed set. My plan is to show that all convergent sequences ${x_n}$ contained in the ...
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Can you have an infinite descending chain of dual spaces?

If a Banach space $X$ is not reflexive, then you have an infinite ascending chain of (continuous) dual spaces: $X’$, $X’’$, $X’’’$, etc. None of these are isomorphic to each other or to $X$. My ...
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1answer
59 views

Are $c_0$ and $c$ duals of some spaces?

The (continuous) dual of a normed vector space is always a Banach space, but the converse is not true. That is, not all Banach spaces are isomorphic to the dual space of some normed vector space. ...
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1answer
25 views

Describe a plane in a matrix metric space

I have a matrix metric space where the distance between two matrices is given by the Frobenius metric. Distance between matrices $A$ and $B$: $$ \sqrt{Tr \left((A-B)^\dagger(A-B)\right)} $$ How do I ...
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67 views

Simpler Proof for the Special Case

Suppose $1 \le p < \infty$ and $f_n ,f \in L^p[0,1]$. If $f_n \to f$ almost everywhere, then prove that $||f_n-f||_p \to 0$ iff $||f_n||_p \to ||f||_p$. The above problem does not come the real ...
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Lemma 2.4.1 Introductory Functional Analysis - Kreyszig :Confirmation of my understanding

I see that questions relating to this lemma have been asked here and here . But my question is quite different in the sense , that i believe that i have understood what the author intends to say but ...
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1answer
38 views

To what extent are measure, topology, distances / norms related?

I m learning about measure theory and I m struggling to make bounds between those different topics. In fact, in topology we saw that a distance induces a topology. But it appears that a measure is ...
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Qutient Topologies in Banach Spaces

I hope the title is not misleading. I am currently reading a paper, where the author uses the following argument: First, let $ f:A\rightarrow B $ be a linear and continuous map between Banach spaces $...
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1answer
58 views

Does the series corresponding to a Cauchy sequence **always** converge absolutely?

Let $X$ be a normed vector space and consider a Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ in $X$. Is it true that the corresponding series of our Cauchy sequence, $\sum_{i=1}^\infty x_i$, always ...
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110 views

If $f$ is integrable, then $\| f\|$ is also integrable.

As usual, a partition of a compact interval $[a, b]$ is, by definition, an strictly increasing family $\Pi = (t_k)_{k = 0}^m$ ($m \geq 0$) of points in the interval such that $t_0 = a$ and $t_m = b;$ $...
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29 views

Show that graph of operator with adjoint operator is closed

Let $X,Y$ be inner-product spaces. Let $T\in L\left(X,Y\right)$ be a linear operator with adjoint operator $S\in L\left(Y,X\right)$ such that $$\langle Tx,y\rangle_Y=\langle x,Sy\rangle_X\quad\forall (...
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23 views

Triangle Inequality for p-norm on nxn real matrices

Consider the following norm: $$ ||A||_F = \left( \sum^m_{i=1} \sum^n_{j=1} |A_{ij}|^p \right)^{1/p}. $$ I would like to prove that it is indeed a norm by proving the triangle inequality: $$||A+...
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1answer
30 views

Rudin Functional analysis, theorem 4.12 corollary (b)

Suppose $X$ and $Y$ are Banach spaces, and $T \in \mathcal{B}(X,Y)$ Then $$ \mathcal{N}(T^*) = \mathcal{R}(T)^{\perp} \;\;\text{and}\;\; \mathcal{N}(T) = ^{\perp}\mathcal{R}(T^*) $$ The corollary ...