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

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inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
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112 views

Trouble in proving that $\|x\|_p = \max|x_j|$

We define p-norm in this way: $\|x\|_p = \{\sum ^N_j=_1|x_j|^p\}^ {1\over p}$ We know that It change to $\|x\|_p = \max|x_j| $ when $ p \to \infty $ How can I prove this ?
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Convex Optimization Closed Form Solution

I'm currently studying for my exame in convex optimization. It covers problem formulation, first and second order conditions of optimality, unconstrained optimization, constrained optimization with ...
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2answers
25 views

spectral radius

Does the spectral radius of a matrix defines a norm? I mean does it satisfy the properties of norm, ie. $$||x|| \ge 0$$ $$||x|| = 0 \implies x=0$$ $$||kx|| = |k|\;||x||$$ $$||x+y||\le ||x||+||y||$$
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show the inequality holds for the matrix relation

How do I choose examples where this inequality holds for the euclidean and infinite norm? $$\frac{1}{||A^{-1}|| \; ||A||} \frac{||r||}{||b||} \le \frac{||e||}{||x||} \le ||A|| \; ||A^{-1}|| ...
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1answer
12 views

Equality in definition of dual space norm

In the definition of the dual space norm, the WP page makes the following statement: and I was wondering why going from the middle equality to the right equality was obvious?
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1answer
1k views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
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58 views

Cauchy Schwarz inequality and absolute value

Here's the inequality: $$|\langle u,v\rangle|\le\|u\|\cdot\|v\| $$ Why on the LHS there's an absolute value? We know that $\langle u,v\rangle \ge 0$ Isn't it redundant?
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25 views

Why $||y|| = \max_{||b|| \leq 1} y^tb$?

I read Application of Legendre transformation in computer vision And at part 5.1 I found a strange equality $||y|| = \max_{||b|| \leq 1} y^tb$ Can anyone provide me intuition why this equality is ...
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1answer
66 views

Norm of two operators

(1) $U(x)=a \langle x,b \rangle +b \langle x,a \rangle $, $a,b\in H\setminus \{0\}$. U is an operator from H to H and a,b are orthogonal elements. I want to calculate $||U||$ For this one I tried the ...
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41 views

Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero.

Let $V$ be an inner product space. Show that if $||x+y||=||x||+||y||$, then $ax=by$ where $a,b$ are non-negative and not both zero. I know that the converse is true. I considered the square of the ...
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1answer
42 views

Functional analysis: $\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}$

Let $(X,\langle\cdot ,\cdot\rangle)$ an inner product space and $A\in\mathcal L(X)$. I have to show that $$\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}.$$ The fact that $\|A\|\geq ...
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23 views

Function to turn results from a nearest-neighbour function into an inversely proportional version?

Short version: Given an input vector D of n values, what are the different methods that one can use to return a vector W such that each value in W is in inverse proportion to the magnitudes of the ...
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8 views

Norm $p$-adic vector spaces

Are there analogs of euclidean norms such as $\infty$-norm in $p$-adic spaces? What are some of analogies between euclidean space and $p$-adic spaces?
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2answers
334 views

Derive steepest descent vector subject to norm constraint

I am currently working through an old textbook Practical Optimization by Gill, Murray and Wright (c 1982) who make some derivations which seem correct, but I am unable to duplicate. In the equations ...
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45 views

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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4answers
58 views

Cauchy-Schwarz, how does it work in this example?

Consider $\|\cdot\|_2$ such that $\|x\|_2 = \left(\sum_{i=1}^n |x_i|^2\right)^{1/2}$. Let $A \in \mathbb{R}^{n\times n}, x\in \mathbb{R}^n$, then $$\begin{align} \|Ax\|_2^2 & = \sum_{i=1}^n ...
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2answers
57 views

Proving a theorem about Fourier coefficients

I need to prove this: Let $f$ be a $C^1$ function on $[-\pi, \pi]$. Prove that the Fourier coefficients of $f$ satisfy $|a_n| \leq \frac{K}{n}$ for some constant $K$. Can someone please let me ...
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1answer
25 views

How to show a norm identity of a weighted sum

I ran across the following identity while reading up on norms. It deals with the square of the $2$-norm of a convex combination. That is, for all $x,y,\in\mathbb{R}^{n}$ and $\rho \in [0,1]$: ...
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taking a tricky limit $\lim_{p\rightarrow \infty} \int_{\mathbb R^N} \left( \frac{|\nabla u|}{\|\nabla u\|_p}\right)^{p-2} \cdots $

$$\lim_{p\rightarrow \infty} \int_{\mathbb R^N} \left( \frac{|\nabla u|}{\|\nabla u\|_p}\right)^{p-2} \frac{\nabla u}{\|\nabla u\|_p}\cdot \nabla v dx \quad\quad \star$$ where $|\cdot|$ is the ...
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4answers
66 views

$\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ is always bounded

Given the matrix $A= (a_{i,j}) \in M_{n,n}$ $||A||=\sup\limits_{x\in X}\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ where $|| $ . $|| _n$ is $ R^N$ norm $\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ is always ...
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56 views

Continuity of norm. Need to understand how and why

$f:X \to \mathbb R \ \ \ , \ f(x)=\| x\|.$ Prove that $f$ is continuous. I have this definition of continuity in metric spaces: Let $(X, d_x)$ and $(Y,d_y)$ be metric spaces. $$f\in C(a) ...
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30 views

Are invertible linear operators of bounded linear operators also bounded?

I have this definition in my book: Definition: Let X,Y be normed linear spaces. An operator $T \in B(X,Y)$ is said to be invertible if there exists $S \in B(Y,X)$ such that $ST=I_X, TS=I_Y$, ...
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17 views

When to use which condition number? (which norm)?

The condition number is used to determine how sensitive b is to changes in A in the equation ...
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2answers
53 views

Norm of the integral operator in $L^2(\mathbb{R})$.

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(y)dy$$ To find $\|A\|$ we can use the unitary Fourier transform $F$, ...
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0answers
19 views

How to show the equivalence of an Euclidean norm to Euclidean distance?

I have the following problem: I have a set of $N$ vectors. I have some a priori information where the information for each vector comes from. If the two vectors $x_1$ and $x_2$ have been sampled ...
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1answer
346 views

Finding matrix norm equivalence constants

I've been given the following: "Find the best positive constants $\alpha$ and $\beta$ such that $\alpha\left\|A\right\|_2\leq\left\|A\right\|_1\leq\beta\left\|A\right\|_2$ for all ...
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1answer
22 views

Prove continuity for a given norm

I struggle with this exercise from an analysis 2 book I use for self study: Let V := $C^1([0,1]; \mathbb{C})$ the vector space of continously differentiable functions from $[0,1]$ to $\mathbb{C}$ ...
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30 views

How can I solve the following exercise

Prove that a linear operator $T:X\rightarrow Y$ is bounded if and only if it maps sequences that converge to zero to bounded sequences .
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62 views

measure theory problems and step functions

I have several questions that I haven't worked out. Any hints or solutions will be appreciated. Suppose that {$f_n$} is a sequence of real valued continuously differentiable functions on [$0,1$] ...
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1answer
34 views

Apply Cauchy-Schwarz to vector?

For $x_i, y_i \in \mathbb{C}$, the C-S inequality gives $$\left| \sum_{i=1}^n x_i \bar{y}_i \right|^2 \leq \sum_{j=1}^n |x_j|^2 \sum_{k=1}^n |y_k|^2 .$$ Is it true if $x_i, y_i$ are actually ...
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0answers
33 views

How to find derivative of $\left\Vert x-a\right\Vert ^{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}$?

Here is what I've tried: $\alpha\left(x\right)=\sum_{i=1}^{n}\left(x_{i}-a_{i}\right)^{2},\ \ \beta\left(y\right)=\sqrt{y}, \ \ \ \gamma\left(z\right)=z^{p}$. Clearly: $\left\Vert x-a\right\Vert ...
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Derivative of Norm of a difference of vectors

I have an expression: $||ax-b||^{2}_{2}$ where both $x$ and $y$ are vectors. I want to find $\frac{d}{dx}||ax-b||^{2}_{2}$, which is the vector derivative of the norm wrt to the vector x. Does ...
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17 views

How to use the condition number to determine whether a matrix is easily inverted?

I have an enormous covariance matrix, but I don't know if it is feasible to take its inverse. So, I thought about finding its condition number, which would help give insight into how easy it might ...
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Is the $L_\infty$ norm of the $\mathcal{l}_2$ norm of this sequence of functions finite?

I am interested in proving or disproving the following claim and am stuck. We define a series of functions with the following properties. For each $i\in \mathbb{N}$ let $f_i\colon \mathbb{R}^+ \to ...
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382 views

Is there a name for this type of vector norm?

In the case of the $\mathcal{l}_2$ norm we have, $$||\mathbf{x}||_2^2=\mathbf{x}^T\mathbf{x}.$$ I was wondering if there was a type of norm that had a linear operation embedded in it, like this, ...
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60 views

Is this a Norm?

How is the following formula calculated, assuming $a$ and $b$ are $n$-dimensional vectors? $\parallel \overrightarrow{a} - \overrightarrow{b}\parallel^2$
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70 views

Calculating Cosine Similarity with Matrix Decomposition (matrix multiplication with normalized columns)

To calculate the column cosine similarity of $\mathbf{R} \in \mathbb{R}^{m \times n}$, $\mathbf{R}$ is normalized by Norm2 of their columns, then the cosine similarity is calculated as ...
3
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354 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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271 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 ...
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comparison of norms in vector ODEs

Assume I have the following equation $$ \dot{f}(t) \le A(t)f(t),\quad f(0) = f_0$$ where $f:[0,\infty]\rightarrow\mathbb{R}^n$ has positive components ($\ge0$), ...
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Prove the following using singular value decomposition

Let $\mathbf{C}\in\mathbb{R}^{m\times n+1}$ have the following singular value decomposition \begin{equation} \mathbf{C} = \mathbf{U}\boldsymbol\Sigma\mathbf{V}^T\\ \mathbf{V} = ...
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43 views

Does spectral norm of a square matrix equal to its largest eigenvalue in absolute value?

I have one simple question. Given the spectral norm $\left \| . \right \| _2$ of a matrix $A$, which is equal to the squareroot of the largest eigenvalue of $A^{^*}A$ $$\left \| A \right \| ...
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1answer
60 views

Can I solve a problem like a combination of PCA and compressed sensing?

$$ \underset{A,x}{\text{minimize}} \quad \lambda \left\| x \right\|_{1} + \left\| A \right\|_{*} $$ $$ D = A + Mx $$ Where $M \in \mathbb{R}^{n \times m}$, $x \in \mathbb{R}^{m \times z}$, $E=Mx \in ...
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1answer
36 views

Can anyone help out with this proof, certain steps are unclear. Norm of linear operator.

I have the following norm defined as follows (in $R^n$, $x=(x^1,x^2,\ldots,x^n)\ $)$\| x\|_1= \sum_{i=1}^{n}|x^i|$ Let $A:R^m \to R^n$ a linear map of the spaces $(R^m ,\| \cdot \|_1 )$ and $((R^n ...
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Matrix Norms Inequality Proof

How do I prove this inequality / equivalence about matrix p-norms? It appears on the wikipedia and mathworld.wolfram pages on matrix norms without proof. $\|A\|^2_2 \leq \|A\|_1 \|A\|_\infty$ Maybe ...
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1answer
43 views

How to computer the proximal operator of a indicator function?

For $G_{1}(x) = I_{c}(x), c =\{x|Ax=b\}$, the Proximal operator is: $Prox_{\gamma G_{1}}(x) = Proj_{c}(x) = x + A^{T}(AA^{T})^{-1}(y-Ax)$ I hope to know, how to derivative this result. And, for ...
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1answer
16 views

Comparison of the norms of two non-negative real-valued vectors differing only in one component.

Let $0 \leq a \leq b$ and let $\mathbf{x} \in \mathbb{R^{n}}$. Let $\|.\|$ be a norm over $\mathbb{R}^{n+1}$. If we write $( \mathbf{x},a)$ the vector of $ \mathbb{R}^{n+1}$ made by concatening ...
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1answer
21 views

Can varying $p$ in the $p$-norm induced distanced change which pair of points are closer?

i.e. For some $x, y, z \in \mathbb{R}^{n}$, do there exist $p_{1}, p_{2}$ s.t. $ 0 < \sum_{i=1}^{n} (|x_{i} - y_{i}|^{p_{1}} - |x_{i} - z_{i}|^{p_{1}})$ and $ 0 \ge \sum_{i=1}^{n} (|x_{i} - ...
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1answer
15 views

Bound on maximum of product of matrix and vector

I need to bound the absolute maximum of each entry of a matrix-vector product: $\max_{|x|_{1}=1} |Ax|_{\infty}$ I tried to pose this in terms of the induced infinity norm of $A$, as in ...