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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49 views

Prove that a set $G$ is equal to $\operatorname{int}(\Omega)$ given that $G \subset \Omega$

given $ \Omega \subset \mathbb{R}^{n} $ an open set and given $ G \subset \mathbb{R}^{n} $ , also open, such that $ \operatorname{int}(\Omega) \subset G \subset \Omega$. Prove that $G= \operatorname{...
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Infinite bounded set in Topological Vector space [closed]

Let X be any topological vector space over the field K then show that there exists an infinite set B which is bounded in X.
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Functional Analysis : completion of normed spaces

The question says: Let be $N$ a normed space with $\dim(N)<\infty$. Suppose that exists a subset $X$ of $N$ such that $X$ has an opened subset $U$. Show that $X$ isn't compact. This question is ...
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If $\{x_n=y_n + z_n\}$ converges, do $\{y_n\}$ and $\{z_n\}$ converge in Banach space?

Let $(X, ||\cdot ||)$ be a Banach space such that $X=Y \oplus Z$, where $Y$ and $Z$ are closed subspaces of $X$, such that every $x\in X$ can be uniquely written as $x=y+z$, $y\in Y, z \in Z$. Then $Y$...
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38 views

Prove the triangle inequality only using the inverse triangle inequality.

I know the easy way to show the inverse triangle inequality using the triangle inequality. As far as I‘m aware of the two are equivalent so I was wondering how to prove the triangle inequality using ...
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1answer
37 views

Show that if $T:E\times F\rightarrow G$ is a non-identically zero bilinear map, then there's two sequences with this property [closed]

Let $E,F,G$ be normed spaces, and let $T:E\times F\rightarrow G$ be a non-identically zero bilinear map, then there are two sequences $(u_n)_{n\in\mathbb{N}}$ and $(v_n)_{n\in\mathbb{N}}$ in $E\times ...
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How do I prove that the infinite union of this specific family of closed sets in $\mathbb R^2$ is neither open nor closed

We observe the vector space $\mathbb{R}^2$ with the Euclidean Norm, for each $k \in \mathbb{N_0}$ we define the set $M_k$ as follows: $$M_k=\Bigg\{x\in \mathbb{R^2}\bigg|\Bigg\|x- \begin{pmatrix} \...
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Visualizing the norm of a bounded linear functional

$\def\b{\mathbb}\def\F{\b F}\def\R{\b R}\def\C{\b C}\def\n#1{\|#1\|}\def\abs#1{\left|#1\right|}$Setup: Let $X$ be a vector space over a field $\F$. (For simplicity, let $\F \in \{\R,\C\}$.) Let $\n\...
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On eigenvalues of compact operators on normed spaces

I'm trying to find a counterexample for this theorem (V.Moretti, "Spectral Theory and Quantum Mechanics", 2nd ed.): Let $X$ be a normed space and $T ∈ B_∞(X)$. $ \forall \ \delta > 0$ ...
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Show that expression $f_n(t)=\int ^t_0 ne^{n(s-t)}f(s) ds$ such that $\lim_{n\to \infty} \|f_n-f\|_{L^p}=0$

Fix $p\in [1,+\infty)$ and let $f\in L_p[0,T]$. show that the expression $$f_n(t)=\int ^t_0 ne^{n(s-t)}f(s) ds\quad t\in[0,T], n\in \mathbb{N}$$ defines a sequence of continuous functions such that $\...
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1answer
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Show that a (not identically zero) billinear map $T: E \times F \rightarrow G$ is not uniformly continuous

Let $E,F,G$ be normed spaces, and let $T:E\times F \rightarrow G$ be a bilinear map (not indentically zero). Show that T is not uniformly continuous. As usual, we are using the product norm $\lvert\...
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Approximation of continuous functions vanishing at a point with polynomials vanishing at the same point

Let $C[0, 1]$ denote the set of all real-valued continuous functions on $[0, 1]$. Consider the normed linear space $$ X=\{f\in C[0,1]| f(\frac{1}{2})=0\} $$ with the sup-norm $$||f|| = \sup\{|f(t)| : ...
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Bounded linear operator on Banach space $C[0,1]$.

Question : Let $C[0,1]$ be the Banach space of continuous functions on $[0,1]$ with supremum norm . Discussed about boundedness of the operator $T$ on $C[0,1]$ and it’s inverse, where $T$ is given by $...
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find norm $T:C[0,\pi]\to C[0,\pi]$ by $(Tx)(t)=\int ^t_0 \cos(t-s)x(s) ds$ [closed]

Find the norm of following opeartor $$T:C[0,\pi]\to C[0,\pi] \text{ by } (Tx)(t)=\int ^t_0 \cos(t-s)x(s) ds$$ \begin{align} \| Tx\|_{\infty }&=\sup _{t\in[0,\pi]} \int ^t_0 \cos(t-s)x(s) ds\\ &...
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Discontinuous multiplicative Linear Functional on a non complete normed Algebra

I have come across the following proposition in the book "Complete Normed Algebras" by F. F. Bonsall and J. Duncan in section 16 on page. Definition: A multiplicative linear functional on ...
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If $X$ is separable, then so is its subset

Let $(X,\|\cdot \|)$ be a normed space and $M\subseteq X$. Show that if $X$ is separable, then $M$ is separable. My attempt: Since $X$ is separable $\Leftrightarrow$ $\exists E\subset X:$ $E$ is ...
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Compute the norms $\|\epsilon \sin \omega t\|$ and $\|\epsilon \sin \omega t\|_{L^{\infty}}$

The sapce $X=C'[0,1]$ is the normed linear space of all continuous differentiable functions on $I=[0,1]$ with norm defined by: $$\|x\|=\max_{t \in I}|x(t)|+\max_{t \in I}|x'(t)|$$ Compute the norms $\...
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Properties of conorm definition

I`m having trouble in solving a few questions about this problem (problem $5.4$ of Pugh $2nd$ edition) The conorm of a linear transformation is defined as $$m(T) = \inf\left\{ \frac{|Tv|}{|v|}: v \...
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1answer
50 views

Understanding Theorem 12.38 Bruckner's Real Analysis

The following is a theorem from Bruckner's Real Analysis: How the underlined formulas (red and green) hold?
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Proof $L^{p}(X,\mu)$ is a linear space normed with respect to the norm [duplicate]

I am studying Lp-Spaces and Banach-Spaces and the norm for $ p \in [1, \infty) $ is given by $$||f||_{L_{p}}=\left ( \int _{X} ||f(x))||^{p} \right )^{1/p}$$ but i didn't find how to test it. Do you ...
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$C^{\infty}$-Diffeomorphism between a normed space and its open unit ball

On $\mathbb R^n$, the open unit ball is $C^{\infty}$-diffeomorphic to the whole space. But if we replace $\mathbb R^n$ by a normed vector space, does still be true that the open unit ball is $C^{\...
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1answer
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On compatible and natural norms (Exercise 2.7.12b in Kreyszig's Functional Analysis text)

$ \newcommand{\nc}{\newcommand} \nc{\C}{\mathbb{C}} \nc{\F}{\mathbb{F}} \nc{\R}{\mathbb{R}} \nc{\a}{\alpha} \nc{\n}[1]{\left \Vert #1 \right \Vert} \nc{\abs}[1]{\left \vert #1 \right \vert} \nc{\set}[...
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Consequence of Hahn Banach theorem is the statement correct?

The theorem B page 227 of this book :Introduction to Topology and Modern Analysis (G. Simmons, 2003) is stated as: If $N$ is a normed linear space and $x_0$ is a non-zero vector in $N$, then there ...
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$τ: Y \to Χ$ is a continuous map and $A: C(X)\to C(Y)$ is defined by $(Af)(y) = f(τ(y))$. How $||A||=1$?

The following is from Conway's Functional Analysis : If $X$ and $Y$ are compact spaces and $τ: Y \to Χ$ is a continuous map, define $A: C(X) \to C(Y)$ by $(Af)(y) = f(τ(y))$. Then $A \in \mathcal{B} (...
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Inclusion relation between norm spaces

Let $1\leq p\leq \infty$. For norms $||~~||$ on $\Bbb{K^n}$ defined by $||x||_p=(|x(1)|^p+...+|x(n)|^p)^{1/p}$ and $||x||_\infty=\max\{|x(1)|,...,|x(n)|\}$. Show that $$\bigcup_{1\leq p<\infty}\{x\...
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Let $m\in C[a, b]$. Consider on $(C[a, b], ||m||_∞)$ the Multiplication operator $A:C[a, b]\to C[a, b], Af=mf$. prove that $||A||=||m||_∞$.

Let $m\in C[a,b]\;$. Consider on $\big(C[a,b],\Vert m\Vert_{\infty}\big)$ the multiplication operator $\;A:C[a,b]\to C[a,b]\;,\;Af=mf\;$. Prove that $\;\Vert A\Vert=\Vert m\Vert_{\infty}\;$. My try: ...
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Triangle inequality in norm [closed]

Let $X=\Bbb R^3$. For $x=(x(1),(x(2),x(3))\in X$, let $$||x||=[(|x(1)|^2+|x(2)|^2)^{3/2}+|x(3)|^3]^{1/3}$$. Then $||~||$ is a norm on $\Bbb R^3$. \begin{align*}||x+y||&=[(|x(1)+y(1)|^2+|x(2)+y(2)|^...
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28 views

If a function is separately differentiable is it diferentiable?

Let $X $, $Y $ and $Z $ normed vector spaces $$f:X \times Y \to Z$$ Such that $f(x, \cdot)$ is differentiable for all $x \in X $ and $f (\cdot ,y)$ is differentiable for all $y \in Y $. Is $f $ ...
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1answer
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Show that the modulus of a complex number is a norm [duplicate]

For a function $\|\cdot\|$ to be a norm, it must satisfy the following: Let $X$ be a vector space over the field $F$. For any $x, y\in X$ and $r\in F$ N1) $\|x\|\geqslant 0$ N2) $\|x\|=0 \rightarrow x=...
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1answer
49 views

Is the Euclidean norm the only norm that admits “non-reflective” isometries?

Let $\|\cdot\|$ be an arbitrary norm on $\mathbb R^n$, and suppose there exists an isometry $T:\mathbb R^n\to\mathbb R^n$ that is not simply a composition of reflections along some of the $n$ axes. ...
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Is the Euclidean norm canonical?

In the spirit of this and this question, I'm interested in the motivation for defining the Euclidean norm in $\mathbb R^n$ to be $\|x\|=\sqrt{\sum_ix_i^2}$. Of course, Euclidean geometry provides a ...
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All possible norms on a finite-dimensional vector space?

Let $X=\mathbb K^n$, where $\mathbb K=\mathbb R$ or $\mathbb C$. I have seen proofs that the functions $$\|x\|_p:=\sqrt[p]{\sum_i|x_i|^p},\qquad p\in[1,\infty]$$ are all norms. (The $p=\infty$ case ...
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1answer
18 views

$C^m(\mathbb{R}^d)|_U$ versus $C^m(\bar{U})$ for $U \subset \mathbb{R}^d$ — restriction versus definition on subset

For $U \subset \mathbb{R}^d$ open, let $\|u\|_{C^m(\bar{U})} = \max_{|\alpha| \leq m} \sup_{x\in U}|D^\alpha u|$ and $C^m(\bar{U}) = \{u \colon U \mapsto \mathbb{R} \mid \|u\|_{C^m(\bar{U})} < \...
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Let $f:\mathbb{E} \rightarrow \mathbb{F}$,where $\mathbb{E},\mathbb{F} $ are normed space,such that:

(I) $f(x+y)=f(x)+f(y). $ (II) $f$ is bounded in $B(0,1)$. Then $f \in L(E,F)$ My attempt: I proved $f (q x) = q x$ for all $x \in E$ y $q \in \mathbb{Q}$. Now I want to prove that $f (r x) = r f (x)$ ...
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How to define the uniform norm on functions of $\mathbb{R}^n$?

I am unsure on how to apply the uniform norm on the following problem. Exercise 2.10 Let $C_0\big(\mathbb{R}^n\big)$ be the closure of the space $C_c\big(\mathbb{R}^n\big)$ of continuous, compactly ...
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Is $\operatorname{Lip}_{\beta}[0,1] $ is closed subset in $\operatorname{Lip}_{\alpha}[0,1]$ for $0 < \alpha < \beta \leq 1$ ??

$\DeclareMathOperator{\Lip}{Lip}$ I already proved that if $0 < \alpha < \beta \leq 1$, then $\Lip_{\beta}[0,1] \subset \Lip_{\alpha}[0,1]$. Recall that we say that $f \in \Lip_{\alpha}[0,1]$ ...
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Exercise 4.4 in Brezis 'Functional Analysis' , $L^{p}$ functions [duplicate]

I have exercise 4.4 from Haim Brezis's book: Let $f_{1},\dots,f_{k}$ be $k$ functions such that $f_{i}\in L^{p_{i}}(\Omega)$ whit $1\leq p_{i}\leq\infty$ for all $i\in\{1,\dots,k\}$, and $\sum_{i=1}^{...
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Can a ball not containing the origin contain its negatives?

My question is a bit more general than I was able to describe in the title: Let $(X,\|\cdot\|)$ be a real normed space and $B_r(x)$ be a ball not containing the origin. That is, $0 < r < \|x\|$....
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Using supremum norm to show uniform boundedness of bounded uniformly convergent sequence

I’m trying to prove that if $f_n \to f$ uniformly, where each $f_n$ is bounded and defined on $E\subset \mathbb{R}$, then there exists $A$ such that $f_n(x)\leq A$ for all $n$. I came up with a proof ...
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Relating Norm and Measure for doing calculus.

In $\mathbb{R}$, integration and differentiation are complementary to each other but as we make our definition more abstract the differ, as to define integration we need a measure space and to define ...
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1answer
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constant distance from extreme points of convex set

Let $(X,\|\cdot\|)$ be a finite dimensional normed space and $A\subset X$ be a convex set with nonempty interior and let $extr(A)=\{e_1,e_2,\ldots e_m\}$ (extr(A)={the set of all extreme points of $A$}...
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1answer
39 views

How to define a 'complete' inner product on $C[a, b]$?

I found the following question in an exam and not able to completely crack it. I paste the question as it was there: Suppose $C[a, b]$ denote the set of all real valued continuous functions on $[a, b]...
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56 views

Why to prove $f\in H^1\mathbb{(D)}$, it is enough to prove that $f\in L^1(|z|=1)$?

Prove that $$\sqrt{\frac{1+z}{1-z}}$$ is in the Hardy space $H^1(\mathbb{D})$, where we take the principal branch of the square root. Answer- let $f(z) = \frac{(1+z)^{1/2}}{(1-z)^{1/2}}$ analytic on $|...
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42 views

$\big\{\sin(ax) + b\;\big|\; a > 0$ and $\;b\;$ belongs to $[c,d]$ where $c,d > 0\big\}$ is compact in the space $\;C\big[0,1\big]\;$?

How to show if $\;\big\{\sin(ax) + b\;\big|\; a > 0$ and $\;b\;$ belongs to $[c,d]$ where $c,d > 0\big\}$ is either compact or not compact in the space $\;C\big[0,1\big]\;$ with the supremum ...
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1answer
31 views

Isolated Schauder Basis for a Normed Space

If $\left(e_{n}\right)_{n}$ is a Schauder basis for a normed space $(V,\|\cdot\|)$ prove that each $e_{n}$ is isolated. I have this question in my exercise sheet and I'm not sure how to prove it. In ...
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1answer
41 views

Let $u:\Omega\subset \mathbb{R}^n\rightarrow \mathbb{R}$ a measureble function. Prove that $u\in L^p(\Omega)$

The function $u:\Omega\rightarrow \mathbb{R}$ is such that $$ \sup\left\{\int_{\Omega}|u(x)|v(x): v(x)\geq 0 \text{ in }\Omega\text{ and }||v||_{L^q(\Omega)}\leq 1 \right\} $$ with $\frac{1}{p}+\frac{...
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34 views

Is there a name for this inner product identity?

Let $V$ be an inner product space. For any pair of vectors $(u,v)$ and any pair of scalars $(\lambda,\mu)$, the following identity is satisfied:$$\left\lVert\lambda u + \mu v\right\lVert^{2} = \left\...
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15 views

Show | ||x|| - ||y|| | < ||x - y|| [duplicate]

I'm still confused as to how I can relate norms with the absolute value function, for this particular problem, I have no idea where to start, any guidance will be appreciated.
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37 views

Hahn Banach and dual space

Let $X$ be a normed space and $X′$ its dual space. If $X≠{0}$, show that $X′$ cannot be ${0}$. Theorem: For an element $x_0$ other than $0$, there is a functional $f^\sim$ such that the norm is $||f||^...
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1answer
31 views

Convergence of a sequence in a norm space

I have a normed space $C([0,1])$ i.e. space of continuous function on [0,1] with norm $$\lVert f \rVert _\infty = \sup_{x\in [0,1]} \mid f(x) \, \mid $$ My sequence is $$\sum _{n\ge1} {(-t)^n \over n}$...

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