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

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Counterexamples of Arzèla Ascoli theorem for non-obeyed criteria

I had an exam on functional analysis some time ago, and one of the questions I couldn't make any sense out was the following: Let $\Omega\subset \mathbb{R}$ and $\{f_n\}$ a sequence of continuous ...
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730 views

Calculating the norm of an element in a field extension.

Given a number field $\mathbb{Q}[\beta]$, where the minimal polynomial of $\beta$ in $\mathbb[Z][x]$ has degree $n$, I would like to calculate the norm of the general element ...
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How to find x so that $\|A x\| = \|A\| \|x\|$ holds

The subbordinance property of matrix-vector multiplication states that $\|A x\| \le \|A\| \|x\|$ where $\|x\|$ is the norm of vector $x$ and $\|A\|$ is the induced norm of matrix $A$. Many textbooks ...
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399 views

Does this cross-product norm inequality hold?

Let $\times$ denote the cross-product. $\;$ Is it the case that For all unit vectors $\:\mathbf{x}\hspace{.01 in},\hspace{-0.03 in}\mathbf{y}\hspace{-0.03 in},\hspace{-0.02 in}\mathbf{z}\:$ in ...
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297 views

Help me understand this proof (showing that something is a norm).

I am reading the following paper: Takáč, Peter On the Fredholm alternative for the p-Laplacian at the first eigenvalue. Indiana Univ. Math. J. 51 (2002), no. 1, 187–237. I need help to understand the ...
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214 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
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Monotone matrix norms

[Ciarlet 2.2-10] Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if ...
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940 views

Why do we need semi-norms on Sobolev-spaces?

I have been studying Sobolev spaces and easy PDEs on those spaces for a while now and keep wondering about the norms on these spaces. We obviously have the usual norm $\|\cdot\|_{W^{k,p}}$, but some ...
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Is there a geometric meaning of the Frobenius norm?

I have a positive definite matrix $A$. I am going to choose its Frobenius norm $\|A\|_F^2$ as a cost function and then minimize $\|A\|_F^2$. But I think I need to find a reason to convince people it ...
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What is the reason norm properties are defined the way they are?

Suppose we have a complex vector space $V$. A norm is a function $f : V \rightarrow \mathbb{R}$ which satisfies (i) $f(x) \ge 0$ for all $x \in V$ (Positivity - Non-Negativity) (ii) $f(x + y) \le ...
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why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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Why does spectral norm equal the largest singular value?

This may be a trivial question yet I was unable to find an answer: $$\left \| A \right \| _2=\sqrt{\lambda_{\text{max}}(A^{^*}A)}=\sigma_{\text{max}}(A)$$ where the spectral norm $\left \| A \right ...
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Why is the operator norm of a diagonal matrix it's largest value?

I read this in my textbook have tried working through it - I keep getting max 2-norm(Ax), which is just the magnitude of Ax. How should I do this proof? (note, this is not for homework, I'm just ...
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Proof of Matrix Norm (Inverse Matrix)

Show for any induced matrix norm and nonsingular matrix A that $$ \left\|A^{-1}\right\| ≥ (\left\|A\right\|)^{-1} $$ where $$ \left\|A^{-1}\right\| = ...
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Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
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Is there a nowhere differentiable norm on $\Bbb R^n$?

Is there a nowhere differentiable norm on $\Bbb R^n$? The non differentiable norms that I know (e.g. the one norm $\|\cdot\|_1$ and the infinity norm $\|\cdot\|_{\infty}$) are non differentiable ...
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162 views

Calculate $\left\Vert \begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} \right\Vert$

With $$\left\Vert A \right\Vert=\max_{\mathbf{x}\ne 0}\frac{\left\Vert A\mathbf{x}\right\Vert }{\left\Vert \mathbf{x}\right\Vert }$$ and $$A=\begin{bmatrix}1 & 2\\ 2 & 4 \end{bmatrix} $$ ...
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Why should I avoid the Frobenius Norm?

I vaguely remember the Frobenius matrix norm ( ${||A||}_F = \sqrt{\sum_{i,j} a_{i,j}^2}$ ) was somehow considered unsuitable for numerical analysis applications. I only remember, however, that it was ...
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What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| ...
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Is it true that $A \geq B$ implies $\|A\|_2 \geq \|B\|_2$ for $A,B \geq 0$?

All matrices are real and not necessarily symmetric. Denote by $A \geq B$ the condition that $(A-B)$ has eigenvalues with non-negative real parts. Denote by $\| \cdot \|_2$ the $L_2$ matrix norm. Is ...
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equivalence of norms

I would like a little help here: I have two defined norms over $C^{1}([0,1])$ : $\| A(f)\|=|f(0)|+\max_{x\in[0,1]}{|f'(x)|}$ $\| B(f)\|=\int_0^1|f(x)|dx+\max_{x\in[0,1]}{|f'(x)|}$ I already ...
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156 views

A metric on $\mathbb{R}^n$ such that $d(\lambda x, \lambda y)=|\lambda| d(x,y)$ which is not induced by a norm

Let $V=\mathbb{R}^n$. Let $d:V \times V\rightarrow \mathbb{R}$ a metric on $\mathbb{R}^n$. Assume that for any $x,y\in V$ and $\lambda \in \mathbb{R}$, we have $d(\lambda x, \lambda y) = ...
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a conjecture on norms and convex functions over polytopes

Suppose one has a convex, bounded polytope P $\subset R^n$ and a strictly convex function $f$ defined everywhere on $R^n$. $f$ has a unique minimum; and suppose this minimum occurs somewhere strictly ...
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71 views

Missing a necessary power in this proof - please help.

This question is somewhat related to Gradient Estimate - Question about Inequality vs. Equality sign in one part. That question was related to part (c) of a problem I am working on, and this question ...
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43 views

Integral with absolute value of the derivative

I'm trying to estimate this integral $\int_0^1 t |p'(t)|dt$ using this value $\int_0^1 |p(t)|dt$; here $p $ is a real polynomial. This means, I am looking for an $M>0$ such that $$\int_0^1 |t ...
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200 views

$\exp(\ln(x))=x$ and $\ln(\exp(y))=y$.

Let $(A,1_A,|\cdot{}|)$ be a unital Banach algebra, for instance $A=M_n(\Bbb R)$ or $M_n(\Bbb C)$. What is the union of all open unit balls $B_{\|\cdot{}\|}$ where $\|\cdot{}\|$ ranges over all ...
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What are some uses for other norms on $\mathbb{R}^n$

We all know and love the standard $1,2,$ and $\infty$-norms on $\mathbb{R}^n$. However, I have never seen anyone mention uses for any of the other $k$-norms that I'm defining as ...
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Why is the 2 norm “special”? [duplicate]

Out of all the vector norms, the $2$ norm, or the Euclidean norm, seems to be "special". Primarily, I say this because we use the 2 norm as a means of determining the distance from one point to ...
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Matrix Norm set

I need help with this problem: Let $\|\cdot\|$ and $\|\cdot\|^{\prime}$ two matrix norms, and consider the relation $$\|\cdot\| \leq \|\cdot\|^{\prime}\ \Leftrightarrow\ \|A\| \leq \|A\|^{\prime},$$ ...
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82 views

Metric on $\mathbb R^2$ in which each sphere B((x,y)r) is an equilateral triangle

Is there a metric on $\mathbb{R}^2$ in which each sphere $B\bigl((x,y),r\bigr)$ is an equilateral triangle centered at $(x, y)$, one of the vertices has the form $(x',y)$ with $x'\geqslant x$? ...
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Finding a norm making a subspace dense

Suppose $V$ is a (real or complex) vector space and $W$ is a subspace of $V$. Under what conditions is there a norm on $V$ making $W$ a dense subspace of $V$? That $V$ and $W$ have the same ...
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Proof that $\|(a,b)\| \leq \|(c,d)\|$ if $0 \leq a \leq c$ and $0 \leq b \leq d$ [duplicate]

Possible Duplicate: Is norm non-decreasing in each variable? Let $\| \cdot \|$ be any norm on $\mathbb{R}^{2}$. Let $0 \leq a \leq c$ and $0 \leq b \leq d$. Show that $\|(a,b)\| \leq ...
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Why are we defining the norms on certain vector spaces the way they are?

What's the intuition behind defining $\|x\|_{\infty} = \max_{1 \le i \le n}\{|x_i|\}$ on the space of ordered $n$-tuples of complex numbers? I'm asking because I've been asked to find a norm on the ...
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Norm of a symmetric matrix?

Say I have a symmetric matrix. I have the concept of 2-norm as defined on wikipedia. Now I want to prove (disprove?) that the norm of a symmetric matrix is maximum absolute value of its eigenvalue. I ...
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Is $f(x) = |\arctan(x)|$ a norm on $\mathbb{R}$?

Is $f(x) = |\arctan(x)|$ a norm on $\mathbb{R}$? Im checking if the properties of a norm holds for $f(x) = |\arctan(x)|$. $1. \ f(x) \ge 0 \Leftrightarrow |\arctan(x)| \ge 0 \\ 2. \ f(x)=0 ...
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Does the inner product $\langle \cdot, \cdot \rangle$ induce any other norms other than the 2 norm?

In the lecture my professor wrote that the standard inner product on $R^n$ is given by $\langle x, y \rangle = x^Ty = \sum\limits_{i=1}^n x_i y_i$ which induces a norm $\sqrt{\langle x,x \rangle} ...
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Derivation of the polarization identities?

For a real (or complex) inner product space $V$, the inner product can be expressed in terms of the norm as either $$ \langle x,y\rangle=\frac{1}{4}(\|x+y\|^2-\|x-y\|^2) $$ or $$ \langle ...
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Confused about Euclidean Norm

I am trying to understand that the Euclidean norm $\|x\|_2 = \left(\sum|x_i|^2\right)^{1/2}$ is in fact a norm and having trouble with the triangle inequality. All the proofs I have referred to ...
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Norm of integral operator in $L^1$

What is the norm of the operator $$ T\colon L^1[0,1] \to L^1[0,1]: f\mapsto \left(t\mapsto \int_0^t f(s)ds\right) $$ ?
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Is the max matrix norm induced?

Let $\|A \| = \max_{1 \le i,j \le n} |a_{ij}|$, where $A$ is a square matrix. I can prove that this is a matrix norm, but is it an induced norm?
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Taking a tricky limit $\lim_{p\to\infty}\int_{\Bbb R^N}\left(\frac{\left\lvert\nabla u\right\rvert}{\left\|\nabla u\right\|_p}\right)^{p-2}\cdots$

$$ \lim_{p\rightarrow \infty} \int_{\mathbb R^N} \left( \frac{\left\lvert \nabla u \right\rvert}{\left\| \nabla u \right\|_p} \right)^{p-2} \frac{\nabla u}{\left\| \nabla u \right\|_p}\cdot \nabla ...
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Norm of a linear map is not attained

Prove that the norm of the linear functional $$\phi: l^1 \ni \{x_n \} \rightarrow \sum_{n=1} ^{\infty} (1 - \frac{1}{n} )x_n \in \mathbb{K}$$ equals one but there doesn't exist a sequence $ \{x_n \} ...
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What matrices preserve the $L_1$ norm for positive, unit norm vectors?

It's easy to show that orthogonal/unitary matrices preserve the $L_2$ norm of a vector, but if I want a transformation that preserves the $L_1$ norm, what can I deduce about the matrices that do this? ...
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Show that the norm of the multiplication operator $M_f$ on $L^2[0,1]$ is $\|f\|_\infty$

I'm having some (hopefully small) issues computing the norm of an operator. Firstly, the problem, For $f\in L^\infty[0,1]$, define $M_f: L^2[0,1]\to L^2[0,1]$ by $M_f(g)(x) = f(x)g(x)$. Show that ...
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p-adic norms and products

I came across the following problems about p-adic norms: Problem. Show that $$\prod_{p} |x|_p = \frac{1}{|x|}$$ where the product is taken over all primes $p = 2,3,5, \dots$ and $x \in ...
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181 views

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

Suppose that $ \star: V^2 \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 \| = \| x \| \| ...
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Proving an inequality involving norms of real functions.

If $ f : [a,b] \subset \mathbb{R} \rightarrow \mathbb{R} $ is continuous and differentiable in $(a,b)$, then one can define a norm for such functions as $$ \|f\| = |f(a)| + \max_{x \in (a,b)} |f ...
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Cauchy Sequence some challenge

i read this sentence in one of math books: ‌Every convergent sequence in metric space is a cauchy sequence. would you please some one add more detail, why this is true? thanks.
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1answer
828 views

Why is semi-norm special?

One difference between semi-norm and norm is: "It is possible for $\|v\| = 0$ for nonzero v, $\|\cdot\|$ being semi-norm" I see some papers, and they use semi-norm directly. Why is semi-norm ...
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919 views

Average norm of a N-dimensional vector given by a normal distribution

I'm interested in knowing what is the expected value of the norm of a vector obtained from a gaussian distribution in function of the number of dimensions $N$ and $\sigma$, i.e: $E[||x||_2]$, $x $~$ ...