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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4
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0answers
61 views

$\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$

I'm learning about functions of bounded variations and need to verify my work for this problem: Show that $\| f \|_{BV} = | f(a) | + V_{a}^{b} f$ defines a norm in the space $BV[a,b]$. My ...
2
votes
1answer
22 views

Let $E$ be a normed space and $b\in E$. Then $d(b, \overline{X}) = d(b,X)$.

$d$ is the distance between a point and a set: $d(b,X) = \underset{x\in X}{\inf}\{\|b-x\| \} $ and $X = B(a;r) = \{\|a-x\|<r: x \in E\}$, $ \overline{X} = B[a;r] = \{\|a-x\| \leq r: x \in E\}$ ...
2
votes
1answer
50 views

boundedness of convex functions

Let $X$ be a vector space, $\Omega$ a convex subset thereof and $f:\Omega \to \mathbb R$ a convex function. Then $f$ need not be bounded from below - not even if it is strictly convex, as the example ...
1
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2answers
33 views

Can derivative of a smooth norm be zero?

Let us call a norm on $\mathbb{R}^n$ smooth if its restriction $\| \cdot \|:\mathbb{R}^n\setminus \{ 0 \} \to \mathbb{R}$ is a smooth map. Is it true that its differential at (every non-zero point) ...
3
votes
1answer
50 views

When is the completion of a space of functions a space of functions?

If $V$ is a $\mathbb C$-vector space of functions $f: X \to \mathbb C$ on some common domain $X$ and $\tau$ is a Hausdorff, locally convex topology on $V$, when may the completion of $(V,\tau)$ also ...
0
votes
1answer
13 views

Preservation of the orthonormal operation in a linear transformation [closed]

Let $A: X\rightarrow X$ be a linear transformation, where $X$ is a inner product space. Show that, if $\|Ax\|=\|x\|$ for all $x\in X$, that $(x,y)=(Ax,Ay)$ for all $x,y \in X$. Show also that if ...
3
votes
1answer
59 views

V is isomorphic to U. U is Banach iff V is Banach

$V$ is isomorphic to $U$ as normed vector space. $U$ is Banach if and only if $V$ is Banach. I don't know exactly, it seems easy at first look, but I have deep problem with the way I have to write the ...
-1
votes
2answers
34 views

What is the norm of matrices? Is it related to the norms of linear transformations? [closed]

What are the norms of a matrix? Is there any relation with norm of linear operators/transformations?
0
votes
1answer
84 views

the space of lipschitz function is complete with respect to some norm

Let $V$ be the space of real valued lipschitz functions over $[a,b]$,we define: $M_f=sup_{x\neq y}\frac{|f(x)-f(y)|}{|x-y|}$ and lipschitz norm: $||f||_{Lip}=|f(a)|+M_f$ prove that $V$ with ...
0
votes
1answer
31 views

Equivalency of two norms

Let $U$ be a normed vector space with two norms: $ || . ||,|| . ||^{'} $ For every sequence $\{x_n \}$ that $||x_n-x||\rightarrow 0 $ & $||x_n-y||^{'}\rightarrow 0 $,we can conclude $x=y$. ...
1
vote
1answer
25 views

Let $V$ be an inner product space and $a,b,c \in V$ distinct points. If $\|c-a\| = \|c-b\| + \|b-a\|$ then $c-a = \lambda (b-a)$, $\lambda \geq 1$.

My attempt I called $x = c-b $ and $y = b-a$. Then $\|x+y\| = \|c-a\| = \|c-b\| + \|b-a\| = \|x\|+\|y\|$. The equality holds in the triangle inequality, thus $x= t\cdot y$ or $y= t\cdot x$, $t \geq ...
0
votes
1answer
64 views

Representation of the elements of a normed vector space that has a dense subset.

Suppose that $U$ is a normed vector space and $S\subset U$ is dense.prove that every element of $U $ can be written as an absolutely convergent series of the finite linear combination of the ...
0
votes
2answers
51 views

Prove a sequence in $\ell^\infty$ is bounded but not Cauchy

$e_n$ in $\ell^\infty$ is the sequence whose $n$th entry is 1, and all others are 0. Show that ${e_n}$ from $n=1$ to infinity is bounded but not Cauchy. I'm really not certain of the whole concept of ...
0
votes
0answers
11 views

How to prove: Subset of normed space is bounded iff all bounded linear functionals take finite values on it.

I want to prove the following without using the uniform boundedness theorem. Consider an arbitrary subset B of a normed vector space. Then B is bounded if and only if $\forall f \in X^*$ $ sup\{ ...
0
votes
1answer
50 views

What values of $p$ give convergence to $0$ in $l^p$

Given a sequence $x_n \in l^p$ whose first $n^2$ members equal $\frac {1}{n}$, and all other entries $=0$, for what values of $p$ does the sequence converge to the zero sequence in $l^p$? So do I ...
1
vote
1answer
38 views

Show which of the following norms are equivalent

On the vector space $C^1[0,1]$ of all real valued continuously differentiable functions defined in $[0,1]$, consider the following norms : $\displaystyle ||f||_{\infty}=\sup_{0\le x\le 1}f(x)$ , ...
0
votes
1answer
39 views

What are we allowed to do when we know that two metric spaces are isometrically isomorphic?

What motivated the question Let $E$ and $F$ be two normed vector spaces and let $f: \Omega \subset E \to F$, where $\Omega$ is open. We denote by $\mathcal L(E,F)$ the set $\{L: E \to F, L$ is ...
1
vote
1answer
97 views

If a linear operator is strong-weak continuous, then it is bounded

$X$ and $Y$ are normed spaces and $L: X\to Y$ is a linear operator from $X$ to $Y$. Show that if $L$ is a continuous operator from $X$ with the strong (norm) topology to $Y$ with the weak topology, ...
0
votes
1answer
27 views

How do completeness and being closed differ in a subspace?

I am meant to prove that if $E$ is a finite dimensional subspace of a normed space $X$, then $E$ is a closed subspace. Now I know that if $E$ is a finite dimensional subspace of $X$, then $E$ is ...
3
votes
2answers
48 views

Vector space containing vectors of infinite norm not complete?

Let $V$ be a vector space such that there is a $v \in V$ with $\|v\|_V = \infty$. Can you conclude from this that $V$ is not complete, i.e. that there is a Cauchy sequence in $V$ which does not ...
0
votes
1answer
50 views

Proving that a subspace of a normed vector space is closed

Question: Let X be a normed vector space. If M is a closed subspace of X and x ∈ X − M then M + ℂx is closed where M + ℂx = { y + λ x : y ∈ M , λ ∈ ℂ } There's a theorem from Folland's Real Analysis ...
0
votes
2answers
42 views

Closed set in normed vector space

Is it true that a subspace M of a normed vector space X is closed if the limit of every sequence in M is contained in M? Whether or not X is complete? Are there alternative characterizations of ...
0
votes
1answer
61 views

Does $\|\sum_{i=1}^{\infty}x_i\| \leq \sum_{i=1}^{\infty}\|x_i\|$ hold for a norm?

In a normed linear space $(X,\|\cdot\|)$, by definition we have $\forall x,y \in X$, $$\|x+y\| \leq \|x\|+\|y\|.$$ My question is, is it true (or does the definition of norm imply) that for a sequence ...
1
vote
1answer
28 views

Normed Quotient Space

If $X$ is a normed vector space and $M$ is a proper closed subspace, I want to show that for any $\epsilon>0$ there exists an $x\in X$ such that $\|x\|=1$ and $\|x+M\|\geq 1-\epsilon$. Is there ...
0
votes
1answer
22 views

Normed Vector Space & Compactness

Let $V$ be a real normed vector space. Let $K$ be a compact set from $V$. Show that the set $2K =$ {2$x$: $x \in K$} is also compact. A topological space is compact if every open covering has a ...
1
vote
1answer
39 views

Normed Vector Space

Let $V$ be a real normed vector space. Suppose that $A$ is an open set from $V$. Show that the set $\frac12 A = \left\{ \frac12 x \, : \, x \in A \right\}$ is also open. Let $V$ be a complex vector ...
2
votes
2answers
52 views

Normed algebra with bounded multiplication

Sometimes one finds the following definition of a normed algebra: This is an algebra with a norm on the underlying vector space such that there is a constant $K \geq 0$ such that $|x \cdot y| \leq K ...
1
vote
1answer
26 views

Is there any standard name for this theorem about extension of bounded linear operators in normed spaces without changing the norm?

Let $X$ and $Y$ be normed spaces, both real or both complex; let, in addition, $Y$ be a Banach space; let $V$ be a (vector) subspace of $X$; let $T \colon V \to Y$ be a bounded linear operator; ...
1
vote
2answers
45 views

In a normed vector space X: $x_n \to x$ weakly iff $d(x_n) \to d(x) ~\forall d \in D$, $D$ dense in $X^*$

Good day, I have the following task: Let $(x_n)_{n \in \mathbb{N}}$ be a sequence in the normed vector space $(X, || \cdot || )$ and let $x \in X$. Show that the following are equivalent: (i) $x_n$ ...
2
votes
1answer
33 views

Open mapping theorem for normed abelian groups

A norm on an abelian group is a function valued in $\mathbb{R}_{\geq 0}$ which satisfies $|x|=0 \Leftrightarrow x=0$, $|{-}x|=|x|$, and $|x+y| \leq |x|+|y|$, not necessarily $|z x| = |z| |x|$ for ...
0
votes
2answers
38 views

Exchanging limits with norms and linear functionals

In a normed vector space $X$, when can we say: $\lim\|x_n\|=\|\lim x_n\|$ and further, if $f\in X^{*}$, when can we say: $\lim fx_n=f(\lim x_n)$?
2
votes
1answer
35 views

weak convergence in hilbert space and exchanging of limits

Question: Let $\{x_n\}$ be a sequence of elements of a Hilbert space $X$ which weakly converge to $x\in X$. Assume also that $\limsup\|x_n\|\leq\|x\|$ Show that $\|x_n-x\|\to0$. Proposed Solution: ...
0
votes
1answer
27 views

Find the adjoint operator of an operator on $C \left [ 0,1 \right ]$

Let $\varphi : \left [ 0, 1 \right ] \to \left [ 0,1 \right ]$ be the continuous function. Let $A : C \left [ 0,1 \right ] \to C \left [ 0,1 \right ]$ be an operator defined by $$\left ( Af\right ) ...
2
votes
1answer
34 views

$l^{2}$ completness in the given norm

Let's consider a $l^{2}$ space, equipped with a norm $$||x||_{\infty} = \sup_{n}{|x_{n}|}+\sum_{n=1}^{\infty}{2^{-n}|x_{n}|}$$ I would like to establsh, whether the space is complete in the given ...
1
vote
2answers
31 views

Simple Normed Space Inequality

Let $(V,\|\cdot\|)$ be a normed vector space. Let $x,y,x',y' \in V$. Say I want to estimate $$\left| \|x\|-\|x'\|-(\|y\| - \|y'\|) \right|.$$ Does the following chain of inequalities hold?: ...
2
votes
0answers
82 views

linear problem with $\|.\|_\infty$ and $\|.\|_1$ norm constraints

I have a question regarding a straightforward linear algebra problem, yet the solution is (at least for me) not trivial. Assume the sequences $\phi_i$ with coefficients $\phi_i[n]\in\mathbb{R}$, and ...
0
votes
0answers
52 views

Linear operator between $l^\infty$ and $l^2$

Let $T:\mathcal{l}^{\infty}(\mathbb{R})\to\mathcal{l}^2 (\mathbb{R})$ be given by $$ T\left((x_n)_{n\in \mathbb{N}}\right) \colon= \left(\dfrac{1}{2^n} x_{2^n}\right)_{n\in \mathbb{N}}.$$ Find ...
3
votes
1answer
59 views

Is $C[0,1]$ a manifold?

I know that $C[0,1]$, as a topological space induced by the metric $d(f,g)=\sup_x |f(x)-g(x)|$, is Hausdorff, second countable, and has cardinality same as $\mathbb R$. But is it a manifold? By ...
12
votes
0answers
142 views

Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? ...
2
votes
2answers
63 views

Cauchy Sequence in Normed Space

Let $(E, ||\cdot ||)$ be a normed space and let $(x_n)$ be a sequence in $E$. Show that the following conditions are equivalent: (a) $(x_n)$ is a Cauchy sequence. (b) For every increasing function ...
0
votes
3answers
60 views

The set $A=\{ x :\|x\|=1 \}\subset \mathbb{R}^n$ is compact in $\mathbb{R}^n$ under euclidean norm.

The set $A=\{ x \mid \|x\|=1 \}\subset \mathbb{R}^n$ is compact in $\mathbb{R}^n$ under Euclidean norm. If I can show this is closed and bounded then we are done. It seems bounded but I am not sure ...
0
votes
1answer
24 views

Let $A=\{ f \in \ell^1\, :\,\text{ for each natural number}\, n\,\text{we have}\, |f(n)|<1/2^n \}$. find the closure of $A$ in $\ell^1$.

Let $A=\{ f \in \ell^1\, :\,\text{ for each natural number}\, n\,\text{we have}\, |f(n)|<1/2^n \}$. find the closure of $A$ in $\ell^1$. I know the interior of A is empty. for the case of getting ...
1
vote
1answer
24 views

A is totally bounded in $l_1 $ if and only if $ (e_n ) \in l_1 .$

Let $(e_n)$ be a sequence of positive real number. Show that A ={$ x=(x_n) \in l_1: |x_n| \le e_n (n \in N ) $ is totally bounded in $l_1 $ if and only if $ (e_n ) \in l_1 .$ I have seen similar ...
2
votes
0answers
55 views

Name for the universal normed space associated to a seminormed space

If $(V,p)$ is a seminormed space, then $(V/N,\overline{p})$ is a normed space, where $N=\{x \in V : p(x)=0\}$ and $\overline{p}(x \bmod N) = p(x)$. My question is as follows: Is there a common name ...
1
vote
3answers
387 views

Norms and equivalence classes question

Let $f\in C[0,1]$. Recall two of the norms we considered in class: $$\|f\|_\infty = \sup_{t\in[0,1]}|f(t)|, \quad \|f\|_1 = \int_0^1|f(t)|\ \mathsf dt. $$ Consider the space $C^1([0,1])$ of ...
0
votes
1answer
22 views

Sequence space norm on $\mathbb{R}^2$

I am working on an assignment so I won't be stating the actual question, but it's about showing extensions of a linear functional in $\mathbb{R}^2$ and it says, I quote Determine all it's linear ...
1
vote
2answers
37 views

Prove that any projection on a normed linear on a subspace satisfies $\|I-P\|\geq 1$

Let $M$ be a subspace of a normed linear space $N$. Let $P$ be a (continous) projection on $M$. Then $$\|I-P\|\geq 1 $$
1
vote
0answers
49 views

Hahn-Banach and hyperplane separation

Let's take a look at the following result, based on Hahn-Banach (extension of bounded linear (real-valued) functionals): Let $X$ be a normed space, $U$ a subspace of $X$ and $u_0 \in X$ with ...
0
votes
3answers
31 views

Normed Linear Space ,$p \neq 2$ is $\left \| f\right \|_{p}= \sqrt{f,f}$ for each $ f \in L^P([0,1])$?

For $p \neq 2$, is there an inner product $< ., .>$ on $L^P([0,1])$ such that $\left \| f\right \|_{p}= \sqrt{f,f}$ for each $ f \in L^P([0,1])$? is it true?for P=2 norm is induced by inner ...
1
vote
1answer
36 views

Operator norm: Show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\| \leq 1 , f \in Y^*, \|f\| \leq 1 \}$

Good day, As stated in the title, I have to show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\|_X \leq 1 , f \in Y^*, \|f\| \leq 1 \}$ where $\| \cdot \|$ is the operator norm, i.e. for $X,Y$ vector ...