Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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norm of inverse less than 1

I just wanna ask if what I am doing here make sense: Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
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1answer
222 views

Matrix Norm set #2

As a complement of the question Matrix Norm set and in order to complete the Problem 1.4-5 from the book: Numerical Linear Algebra and Optimisaton by Ciarlet. I have this additional conditions: (3) ...
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1answer
420 views

Poincaré's lemma with norm in $H_{0}^{1}$

I wonder why $\| u\|_{H_0^1} = \int_{\Omega} |Du|^2$ for u in $H_0^1(\Omega)$, with $\Omega = (-1,1)$? I might be wrong but isn't $\| u\|_{H^1} = \| u\|_{L^2} + \| Du\|_{L^2}$? How come that $\| ...
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167 views

Absolute norms and 1-unconditional sums

Absolute norm Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that $$ \|(x,y)\|_N=N((\|x\|, ...
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465 views

$L^p$ norm of multivariate standard normal random variable

Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of $$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$ as $n\rightarrow \infty$? For $p=2$ results about $\chi$-distribution tell us that ...
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1answer
399 views

Prove a basic fact on a linear combination of vectors

Let $x_i (i=1,...,n, n>d)$ be a unit vector in $R^d$. $c_i>0$ is a positive real scalar. How to prove the following fact? Fact: There exist some vectors $x_i$ such that $\sum_{i=1}^n c_i ...
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0answers
73 views

Differentiation operator on smooth function with compact support

Suppose $f$ is $C^\infty$ with compact support. Let $T_n$ be the operator which sends $f$ to its $n$-th derivative. Is $||T_nf||_\infty$ bounded? It seems like I should use Stone-Weierstrass, but ...
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58 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...
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norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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241 views

Effect of doubly stochastic matrix on vector norm

Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector. What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$? In addition if $\Vert x \Vert_2=1$, what ...
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139 views

What are norms used for?

These two questions are quite similar to this one, so I apologise if this irritates anyone. Also, I suspect that a lot can be said in the answer, so I am really just looking for some main points ...
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3answers
646 views

Example of a sequence that converges to two different limits with respect to two complete norms

I've wondered about the following question : Is there an (explicit?) example of a vector space $X$, two complete norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on $X$, and a sequence $(x_n) \subseteq X$ such ...
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4answers
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Defintion of $L_\infty$ norm

Where does the definition of the $L_\infty$ norm come from? $$\|x\|_\infty=\max \{|x_1|,\dots,|x_k|\}$$
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647 views

Normalizing a matrix

I came across a step in an numerical algebra algorithm that says "Normalize the rows of matrix A such that they are unit-norm. Call U the normalized matrix." I do something like this: ...
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3answers
1k views

equivalent norms in Banach spaces of infinite dimension

Suppose $ X $ is a Banach space with respect to two different norms, $ \|\cdot\|_1 \mathrm{ e } \|\cdot\|_2 $. Suppose there is a constant $ K > 0 $ such that $$ \forall x \in X, \|x\|_1 \leq ...
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727 views

How do you prove the $p$-norm is not a norm in $\mathbb R^n$ when $0<p<1$?

I see that it fails to satisfy the triangle inequality by example but I don't see how to prove this is the case for all $0 < p < 1$. The definition I am using for $p$-norm is $$ \|A\|_p= ...
3
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3answers
68 views

whats the difference between $|v|$ and $||v||$?

$v$ being a vector. I never understood what they mean and haven't found online resources. Just a quick question. Thought it was absolute and magnitude respectively when regarding vectors. need ...
3
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1answer
503 views

Frobenius norm is not induced

My textbook says: Frobenius norm defined on $\mathbb{R}^{m,n}$ by a formula: $$\|A\|_F=\sqrt{\sum_{i=1}^n\sum_{j=1}^m|a_{i,j}|^2}$$ when $n>1, \ m>1$ is not induced by any vector's ...
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3answers
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Example of two norms on same space, non-equivalent, with one dominating the other

I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
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1answer
88 views

Does Permuting the Rows of a Matrix $A$ Change the Absolute Row Sum of $A^{-1}$?

For $A = (a_{ij})$ an $n \times n$ matrix, the absolute row sum of $A$ is $$ \|A\|_{\infty} = \max_{1 \leq i \leq n} \sum_{j=1}^{n} |a_{ij}|. $$ Let $A$ be a given $n \times n$ matrix and let $A_0$ ...
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5answers
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Equivalence of norms: $\|x\|$ and $\|x\|_1=\|x\|+\lvert\,f(x)\rvert$

Let $f: X\to \mathbb{R}$ be a linear functional and $\|\cdot\|_1$ is defined as follow $\|x\|_1=\|x\|+|f(x)|$. Prove or disprove $\|\cdot\|_1$ is equivalent to $\|\cdot\|$ iff $f$ is continuous. I've ...
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132 views

Norm of Hilbert matrix is it equal to $\pi$?

Let $A$ be a Hilbert matrix, $$a_{ij}=\frac{1}{1+i+j}$$ We have the following result : $\Vert A\Vert\leq \pi$. I am using the subordinate norm of the euclidean norm i.e. $$ \Vert A\Vert=\sup\{\langle ...
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2answers
197 views

Is $l^2$ norm differentiable at $x=0$?

For $l^2$ norm on $\mathbb{R}^d$, $\frac{d\|x\|_2}{dx} = \frac{x}{\|x\|_2} $, so $d\|x\|_2$ is differentiable wherever $x \neq 0$. is the norm differentiable at $x=0 \in \mathbb R^d$? Thanks!
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411 views

Norm of differentiation operator $Tf(t)=f^{'}$..

Consider $T:C^1[0, 1]\rightarrow C[0, 1]$ given by $Tf=f'$ where $$\|f\|_{C^1}=\|f\|_\infty+\|f'\|_\infty$$ and $\|f\|_\infty=\sup_{x\in [0, 1]}|f(x)|$. How to prove $\|T\|=1$? The inequality ...
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2answers
139 views

Linear Algebra munkres analysis on manifolds question.

If $A$ is an $n$ by $m$ matrix and $B$ is an $m$ by $p$ matrix, then $$ |AB| \leq m|A||B|$$ where $|A| = \max\{|a_{ij}| : i = 1,\ldots,n \text{ and} j = 1,\ldots,m\}$ Attempt: $ |AB| = \max\{| ...
3
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2answers
195 views

Inequality involving norm and inner product

I am stuck proving this trivial inequality: on a real inner product space, $(||x||+||y||)\frac{\langle x,y\rangle}{||x|| \cdot ||y||}\leq||x+y||$ I have tried to square both sides and use the Cauchy ...
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3answers
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Is every normed vector space, an inner product space

Let $V$ be a vector space over $\mathbb{C}$. If $V$ is an inner product space, then $V$ is normed (where the norm is defined as $\|x\|=\sqrt{(x,x)}\,\,$). Now if $V$ is normed, does it follow that ...
3
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1answer
162 views

The trace norm cannot be increased by composing with a unitary operator

$\newcommand{\tr}{\operatorname{tr}}$ I was reading a proof for the statement $|\tr(US)|\leq |\tr(S)|$, for every endomorphism $S$ on a complex vector space $H$ and every unitary operator $U$ on the ...
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2answers
131 views

Geometric Sums in Banach Algebra

Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm. I can show that $||w|| \le ...
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1answer
40 views

Is the trace of a matrix a norm?

If the matrix norm of A is defined as $\|A\|=\sum_{i,j}|Aij|$ then how do I determine if the sum of the diagonal elements, i.e., the trace is a valid norm? I am not really sure how to approach this ...
3
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1answer
82 views

Is $\|\nabla u\|_2$ a norm on $W^{1,2}(\mathbb{R}^N)$

Is $\|\nabla u\|_2$ a norm on $W^{1,2}(\mathbb{R}^N)$? I know when $\Omega$ is bounded, $\|\nabla u \|_2$ defines a equivalent norm on $W_0^{1,2}(\Omega)$ with $\|u\|_{1,2}$. And to me it seems ...
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1answer
250 views

Norm of Prime Ideal

Show that the norm of a prime ideal in a number field $K$ is a power of some prime number, i.e., if $P$ is a prime ideal in $O_K$ for some number field $K$, then $N_\mathbb{Q}^K(P)=p^n$ for some ...
3
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1answer
131 views

How can we replace $\sup$ with $\max$ in definition of subordinate norm for finite-dimensional vector space?

I have been given the definition of a subordinate (operator or matrix) norm: $$\lvert\lvert A \rvert\rvert=\sup_{x\in V}\frac{\lvert\lvert Ax \rvert\rvert}{\lvert\lvert x \rvert\rvert},$$ where $V$ is ...
3
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236 views

How is $L_{2}$ Minkowski norm different from $L^{2}$ norm?

I am reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use $L_{2}$ Minkowski norm (Euclidean) as the distance metric in the feature space ...
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2answers
63 views

Norms of eigenvalues bigger than 1 implies $|Ax|>x$ for all nonzero $x$?

If all the eigenvalues of $A$ (an n by n real matrix) have norms bigger than 1, is it true that $|Ax|>|x|$ for all nonzero $x\in\mathbb{R}^n$? (This is clearly true if $x$ is an eigenvector ...
3
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1answer
74 views

Showing that the zero vector has norm zero

I need to show that this is a property of a norm. I know this is supposed to be straightforward but I am somehow not seeing it. The property is $$\lVert 0\rVert = 0$$ I was trying to use the fact ...
3
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2answers
244 views

Two proofs concerning Hölder's inequality

I am studying functional analysis and I have come across two statements which can be proven by using Hölder's inequality, but I don't know how/why. Reminder: let $a\in l^p$ and $b\in l^q$, then: $$ ...
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1answer
337 views

An inequality involving the norms of symmetric positive definite matrices

Given A and B two real symmetric positive definite matrices is it true that, for some norm $\|.\|$, this inequality holds $$ \|AB-I\| \leq \|A^2B^2-I\| \qquad ? $$
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The openness of the set of positive definite square matrices

Let $\mathbb{R}^{n\times n}$ be the vector space of square matrices with real entries. For each $A\in \mathbb{R}^{n\times n}$ we consider the norms given by: $$ \displaystyle\|A\|_1=\max_{1\leq j\leq ...
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Bounding the $l_1$ norm of a vector

Let $x$ be real vector with $\|x\|_1=x_1+\ldots +x_{2n}$. How to bound from above $(x_1+\ldots+x_n)(x_{n+1}+\ldots+x_{2n})$ by $l_2$ norm of the vector $x$. Of course, using $\|x\|\leq\sqrt ...
3
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1answer
175 views

Proof that $\|\cdot\|_B$ defines a norm

Can you tell me whether my (partial) proof that $\|v\|_B := \inf \{\lambda > 0 \mid \frac{1}{\lambda} v \in B \}$ where $\varnothing \neq B \subset \mathbb R^d$ is open, bounded, $B = -B$ and ...
3
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1answer
125 views

A name for the property $ \| x \star y \| = \| x \| \| y \| $.

Suppose that $ \star: V \times V \to V $ is some binary operation on a vector space $ V $. Should it hold, is there a name for the following property? $$ \forall x,y \in V: \quad \| x \star y \| = \| ...
3
votes
1answer
35 views

Definition of the norm of a bounded linear operator.

I have a somewhat basic but confusing question regarding the definition of the norm for bounded linear operator. Suppose $f$ is a bounded linear operator, that is, there exists $M>0$ such that ...
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2answers
232 views

Proof that the Euclidean norm is indeed a norm

I apologize beforehand for this question. Its embarrassing I know. Anyway, here we go. Recall: $$ \| x \|^2_{\mathbb{R}^2} = \sum^{n}_{i = 1} x^2_{i}$$ How do we prove its a norm? Well if its a ...
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73 views

Inequality between operator norm and Hilbert-Schmidt norm

I have seen the following inequality here but I don't know where I can find a proof for it. Could somebody give me a hint to understand it or guide me to a reference please? $\|AB\|_{HS} \leq ...
3
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1answer
314 views

Why is semi-norm special?

One difference between semi-norm and norm is: "It is possible for ||v|| = 0 for nonzero v, ||.|| being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm better than ...
3
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1answer
184 views

Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but ...
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1answer
37 views

Spectral Norm of $2\times 2$ symmetric matrix

Consider a $2\times 2$ symmetric matrix, in this case, is there some closed formula for its spectral norm ? By spectral norm I mean the induced 2-norm, there is a definition here. Thanks.
3
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113 views

Question about sup norm

Let $x \in \mathbb{R}^n$. Define $|x| = \max\{ |x_1|,...,|x_n|\} $. I want to show that this is a norm on $R^n$. This is my reasoning. First, notice $$ |x| = \max\{ |x_i| \} \geq |x_i| \; \forall i ...
3
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1answer
289 views

Prove that the given distance function is a norm

Let the vector space $X=K^3$. For $x=(\alpha_{1}, \alpha_{2}, \alpha_{3}) \in X$, we define $||x||= [(|\alpha_{1}|^2+|\alpha_{2}|^3)^\frac{3}{2} + |\alpha_{3}|^3]^\frac{1}{3}$ Proof that $||·||$ is ...