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

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p-norm of series / Mathematica

When computing the $p$-norm of the series $x_n = \mathrm e^{\mathrm i n}$ for $n \in \mathbb{Z}$ (not $\mathbb{N}$), the sum should not converge: $$\|x\|_p^p = \sum_{n \in \mathbb{Z}} (\mathrm ...
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P-adic “Norm” and scalability criterion

I just came across the p-adic norm for the first time. I tried to show that it is actually a norm on $Q$ but I was asking myself, whether checking scalability is a bit self referential ? What I mean ...
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156 views

How do you expand this frobenius form?

I'm trying to expand this Frobenius form $||C \circ (A-XB)||_F^2$ (here $\circ$ is the Hadamard point-wise multiplication). I want to find the minimum value with respect to X. $$ ...
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Do real vectors attain matrix norms?

The $p$-norm of a matrix $A$ for $p \geq 1$ is defined as $$\|A\|_p = \max_{ x \in \mathbb{R}^n, \|x\|_p=1} \|Ax\|_p.$$ My question: does this equal $$ \max_{ x \in \mathbb{C}^n, \|x\|_p=1} ...
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275 views

$\|f+g\|_p\leq\|f\|_p+\|g\|_p$ where $\|f\|_p=(\int_{a}^{b}|f(t)|^p dt)^\frac{1}{p}$

Using the fact that $$cd\leq\frac{c^p}{p}+\frac{d^q}{q}$$ if $$\frac{1}{p}+\frac{1}{q}=1$$ and letting $$c=\frac{|f(t)|}{\left(\int_{a}^{b}|f(t)|^p dt\right)^\frac{1}{p}}$$ and ...
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Maximum subset sum of $d$-dimensional vectors

This is a $d$-dimensional generalisation of the post Inequality with Complex Numbers. (See my comment under Robert Israel's answer.) Generalising Potato's proof for $d$-dimensions, we can show the ...
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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= ...
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How does one prove that the spectral norm is less than or equal to the Frobenius norm?

How does one prove that the spectral norm is less than or equal to the Frobenius norm? The given definition for the spectral norm of $A$ is the square root of the largest eigenvalue of $A*A$. I don't ...
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Proof of Neumann Lemma

Prove that if $\|A\| < 1$, then $I-A$ is invertible. Here, $\|\cdot\|$ is a matrix norm induced by a vector norm. This lemma is referred to as Neumann Lemma. Any ideas on how to go ahead with ...
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144 views

A question on norm of error vector

Let $(s_n)_{n \in \mathbb{N}}\in\ell^2(\mathbb{N})$ (i.e. $\displaystyle \sum_{n=0}^{\infty}\vert s_n\vert^2<\infty$). Define vectors $A=[A_1,\ldots,A_M]$ and $B=[B_1,\ldots,B_M]$ with coordinates ...
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Norm with special conditions

Let $N$ be a norm on $\mathbb R^2$, such that $N ( \mathbb Z^2) \subset \mathbb N $, where $\mathbb Z^2 =\{ (a,b)\mid a\mbox{ and }b \mbox{ are integers}\}$. Help me to prove that for $u$, $v$ fixed ...
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How many elements in a number field of a given norm?

Let $K$ be a number field, with ring of integers $\mathcal{O}_k$. For $x\in \mathcal{O}_K$, let $f(x) = |N_{K/\mathbb{Q}}(x)|$, the (usual) absolute value of the norm of $x$ over $\mathbb{Q}$. ...
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217 views

$L_1$ projection of sum of convex functions onto polytopes

Suppose I have a function $f(x) : \mathbb R^n \to \mathbb R$ that is the sum of a given strictly convex function $g : \mathbb R \to \mathbb R$ in a single variable, i.e. $f(x) = g(x_1) + g(x_2) + ...
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797 views

Multiplication operator norm

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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What is the proper term for the entity that relates a vector space and a set?

One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) ...
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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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Convergence of $\frac1m(I+A+A^2+\cdots+A^{m-1})$

Let $A$ be an $n\times n$ matrix of nonnegative entries such that $A_{i1}+A_{i2}+\cdots+A_{in}=1$ for all $i\in\{1,2,\ldots,n\}$. What does $A$ have to satisfy so that the sequence ...
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107 views

Quick question: finite extensions and norms

[Edit: "Suppose K is a field complete with respect to a discrete valuation, with valuation ring $\mathcal{O}$."] I'm trying to solve the following problem: let $L/K$ be a finite extension of fields, ...
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Finding all scalars $k$ such that $\| kv \| = 10$

I have a homework question that asking to Find all scalars $k$ such that $\|kv\| = 10$ when $v=(1,-4,6)$. What I did that that I found the norm of $v$ which I found to be $\sqrt{53}$. Then I ...
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470 views

A question on linear transformation

Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is $$\left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ ...
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122 views

Example of a vector norm for which $\|I\|<1$

In order to prove a larger assumption, I need to find a vector norm over $M_n$ such that $\|I\| < 1$. None of the standard $p$-norms, nor the infinity norm work. I know that for matrix norms, ...
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63 views

Equivalence of a Vector Norm being Absolute

I'm trying to show that a vector norm $\|\cdot\|$ being absolute ($\|x\| = \|\;|x|\;\|)$ is equivalent to showing that $\|x'\| = \|[\alpha_1x_1\ldots\alpha_nx_n]^T\| = \|x\|$ for all ...
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323 views

Matrix Norm Identity Derivation

I am having trouble figuring out where something in a book I am reading is coming from. (The book is Matrix Computations by Golub and Van Loan, 3rd edition, p.58.) It will probably be obvious once ...
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645 views

Cauchy sequence in a normed space

Let $V$ be a real vector space. Suppose that $\Vert \cdot \Vert_1$ and $\Vert \cdot \Vert_2$ are two norms on $V$ which are equivalent. I suspect the following to be true. Let $(x_n)_{n=0}^\infty$ ...
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Convergence of a pair linearly independent elements of a vector space

Let $v_1$ and $v_2$ be linearly independent elements of some normed vector space $V$. We have a sequence $(a_n,b_n)$ and are told that $a_nv_1+b_nv_2$ converges to the zero element. Is it the case ...
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148 views

Help with a proof that the inversion mapping for matrices is continuous

I'm reading a proof that the map $A\mapsto A^{-1}$ is continuous in the operator norm. A part of the proof is that if $A,B$ are matrices such that $A$ is invertible and $\beta=\parallel ...
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870 views

Subordinate matrix norm

I have the following matrix norm: $$\Vert A \Vert = \max_{1\leq i, j\leq n} \vert a_{ij} \vert \>.$$ I have to decide if this is a subordinate matrix norm or not. I have tried to use the ...
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how to prove vector norm equivalence in finite dimensional space($\mathbb{R}^{n}$)?

In most of the vector norm material, it was mentioned that the following inequalities can be proved, but no one provided the proof: $$\lVert x\rVert_2\le\lVert x\rVert_1\le\sqrt{n}\lVert x\rVert_2;$$ ...
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180 views

Maximum 1-norm of matrix times unit vector

If $A$ is a $p \times p$ matrix, what is $$\max_{||u||_2=1} ||Au||_1 ?$$ I am specifically interested in the case when $A$ is positive definite.
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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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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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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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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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Orthogonal in the B Norm?

If you have two generalized eigenvectors $\varphi_1 , \varphi_2$ (with different eigenvalues) of a matrix A, then they will be orthogonal in the B norm. In this context, I do not ...
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Does convexity of a 'norm' imply the triangle inequality?

Given a vector space $V$ (for convenience, defined over $\mathbb{r}$), we call $d:V\rightarrow\mathbb{R}$ a norm for $V$ if $\forall \mathbf{u}, \mathbf{v} \in V$ and $\forall r \in \mathbb{R}$ we ...
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Difference between metric and norm made concrete: The case of Euclid

This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me. This time I am making ...
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282 views

Matrix norm characteristics

$$\|A\|_p = \displaystyle \max_{\|x\|_p = 1} \|Ax\|_p $$ $$\|A\|_2 \leq \|A\|_F \leq \sqrt{n}\|A\|_2$$ How I can show that $1$ and $2$ are correct? $2)$ $||Ax||_{2}=\sqrt{\sum_{i=1}^{n} ...
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Norms for space $V$ of vector-valued mappings $f: \mathbb R^m \rightarrow \mathbb R^n$

I wonder whether there are natural norms for the space $V$ of vector-valued functions that map $\mathbb R^m$ into $\mathbb R^n$. Formally, let's define $V$ as the set of $f$ such that $f: \mathbb R^m ...
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How do I calculate the $p$-norm of a matrix?

I know that the $p$-norm for a matrix is: $$\|A\| = \max_{x\neq 0} \frac{\|Ax\|_p}{\|x\|_p}$$ but I don't know what this really means. So how would I compute the $2$-norm, $3$-norm, etc for the ...
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How to define an inner product for normed space? [duplicate]

Possible Duplicates: Norms Induced by Inner Products From norm to scalar product Given a normed space $(X, \|\cdot\|)$. Is it possible to define an inner product $\langle \cdot, \cdot ...
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What is the difference between the Frobenius norm and the 2-norm of a matrix?

Given a matrix, is the Frobenius norm of that matrix always equal to the 2-norm of it, or are there certain matrices where these two norm methods would produce different results? If they are ...
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Is a completion of an algebraically closed field with respect to a norm also algebraically closed?

Assume we have an algebraically closed field $F$ with a norm (where $F$ is considered as a vector space over itself), so that $F$ is not complete as a normed space. Let $\overline F$ be its completion ...
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Ky Fan Norm Question

How can one simply see that Ky Fan $k$-norm satisfies the triangle inequality? (The Ky Fan $k$-norm of a matrix is the sum of the $k$ largest singular values of the matrix) Thanks.
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parametrize hypersphere

I want to find an $n$-vector $\hat{\theta}$ that maximizes a function $f(\theta)$ subject to the $p$-norm constraint $||\theta||_p = c$. Is there a general parametrization of $p$-norm hyperspheres ...
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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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How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?

I want to prove the following theorem (no idea whether it has a name): Let $V = \mathbb{R}^n$ or $\mathbb{C}^n$ and $\|\cdot\|$ be a norm on $V$. Then, there exist $C_1, C_2 > 0$ such that for all ...
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534 views

Projecting onto vector space with L-Infinity norm/ minimum absolute value polynomial fitting

I'm looking for a way to project a vector (in this case a function on the real line) onto a basis for that space (in this case the set of N-degree polynomials over the domain of a closed interval) ...
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Norms Induced by Inner Products and the Parallelogram Law

Let $ V $ be a normed vector space (over $\mathbb{R}$, say, for simplicity) with norm $ \lVert\cdot\rVert$. It's not hard to show that if $\lVert \cdot \rVert = \sqrt{\langle \cdot, \cdot \rangle}$ ...
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From norm to scalar product

In the Eculidean Space, one can automatically define a norm if a specific scalar product is given. On the contrary, it's not always true. A p-norm is a scalar product if and only if p=2. My question ...
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Understanding the properties and use of the Laplacian matrix (and its norm)

I am reading the wikipedia article on the Laplacian matrix: http://en.wikipedia.org/wiki/Laplacian_matrix I don't understand what is the particular use of it; having the diagonals as the degree and ...