Questions tagged [norm]

This tag is for the questions relating to the norm which is a function on a vector space $X$ that generalizes notion of length of vector in general vector spaces. A vector space $~X~$ with a distinguished norm is called a normed space.

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1answer
27 views

The operator norm $\|L\|$

Let $C_0([0, 1])$be a subspace of $C([0, 1])$, a functional space consisting of real-value continuous functions over the interval $[0, 1]$, such that $C_0 ([0, 1]) = \left\{ f \in C([0, 1]) \mid \...
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1answer
25 views

How $N(2\zeta^{2n})=2^{p-1}$?

Let, $\zeta$ be the $p$ th root of unity and $\mathbb{Z}[\zeta])$ be the number ring generated by $\zeta$, and $N$ is the norm function. Why or how $N(2\zeta^{2n})=2^{p-1}$? The source of the problem ...
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25 views

Pythagorean inequality in l1 norm for projection onto simplex

Let $x$ be a point in $\mathbb{R}^d$, let $z$ be the projection of $x$ onto the $d-1$ dimensional simplex ($z^\top \mathbf{1}= 1$ and $z \succeq 0$). So basically $$z = \arg\min_w \{\rVert x-w\rVert_2 ...
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2answers
34 views

Definitions of the operator norm of a matrix

I understand that the operator norm of a matrix is induced by the vector norm, and is given by $$ \|A\|_{\rm op} = \max_{\|x\| = 1}\|Ax\| $$ However, in a lecture, I see the operator norm being ...
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12 views

Differentiability of $p$-norm and it's $p$-th power at origin.

In Differentiability of Norms, we saw that all $p$-norms are not differentiable at the origin. If you instead take the $p$-norm to the $p$-th power, does this guarantee differentiability everywhere ...
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1answer
15 views

Solving least squares with QR factorization

I'm looking at the notes on https://www.cs.cornell.edu/~bindel/class/cs3220-s12/notes/lec11.pdf. On the first page, we have the following steps \begin{align} ||Ax-b||^2&=||Q^T(Ax-b)||^2\\ &=\...
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1answer
51 views

$A > B$ implies $1 > A^{-1} B$ for operators on Hilbert Spaces

As the title suggests, I have two operators on Hilbert spaces. They are both unbounded but I have bounded the inverse of $A$ (by showing it is strictly positive.) I seek to bound the composition $A^{-...
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1answer
65 views

Bounded Linear Operator from $C_0([0,1])$ to $C([0,1])$

Define $$ C_0([0,1]) := \left\{f\in C([0,1]) : \int_0^1f(t)\, \mathrm dt=0\right\}. $$ Show that $T : C_0([0,1]) \to C([0,1])$, given by $$ (Tf)(x) := \int_0^x(t-x) f(t) \, \mathrm dt, $$ defines a ...
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1answer
12 views

Norm with supremum

hope your having a good day, I have stumbled across this question in Topology, which is to show that: N((x, y)) = Sup|xcost+ysint| For all t belong in [0,2π] and (x, y) in R^2 is a norm. Hooe you ...
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0answers
23 views

How to find all planar squares with respect to the taxicab norm or the uniform norm?

Excuse me if this is a silly question. Let $\|\cdot\|$ be any norm on $\mathbb R^2$. We say that four points $a,b,c,d\in\mathbb R^2$ form a square with respect to the norm $\|\cdot\|$ if $\|a-b\|=\|b-...
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23 views

Generalized norm defined in an infinite algebraic lattice

I am wondering whether it makes sense to define a generalized norm (generalized absolute value) of an element of a (new) concrete finite-dimensional unital algebra $A$ over complex numbers. The unital ...
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0answers
13 views

Relation between $\mathcal{L}_2$-norm of a signal and its amplitude in the frequency domain

Suppose for the signal $x(t)$, we know that $\|x\|_{\mathcal{L}_2}\le 1$, with this norm defined as $\|x\|_{\mathcal{L}_2}:= \sqrt{\int_0^{\infty}\|x(t)\|^2\mathrm{d}t}$, where $\|\cdot\|$ is the ...
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0answers
18 views

Is the “double” total derivative of the norm, the metric?

Let $||\cdot||$ be the norm of a vector $\mathbf{v} \in \mathbb{R}^3$ $$ ||\mathbf{v}||=\sqrt{x^2+y^2+z^2} $$ The total derivative is: $$ d||\mathbf{v}||=\frac{1}{2\sqrt{x^2+y^2+z^2}}(2xdx+2ydy+2zdz) $...
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1answer
19 views

Will $\sqrt{h \sum_{i =0}^{N-1} (1 - u_i^2)^2} < C_1$ imply there exist $C_2$ satisfies $\sqrt{h \sum_{i=0}^{N-1} u_i^2} < C_2$ [closed]

Assume the interval $[a, b]$ is divided by uniform grids $x_i = a + i * h, i = 0, 1, \cdots, N$, where $h = \frac{b - a}{N}$, $\mathbf u = [u_0, \cdots, u_{N-1}]^T$ is a grid funciton, will the ...
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0answers
23 views

Specific matrix norm

I have a real, limited scalar field $S (x, y)$ which I describe with a matrix $M$ with an approximate value of $S (x, y)$ within each cell. The bigger the matrix, the more precise this description ...
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1answer
42 views

Why does the metric on a Schwartz space generate the same topology as the family of seminorms?

I am reading Rauch's "Partial Differential Equations", and he makes a jump I don't understand. He defines the Schwarz space as the space of $C^\infty$ functions that decrease faster than any ...
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1answer
48 views

Householder transformation matrix proof norm [closed]

Definition: Let $x \in \mathbb R^n$ and $Q:=(I-2*ww^T/w^Tw)$ the Householder matrix Exercise: Is the vector $w = x + ∥x∥e_1$ not equal to zero, than is $Qx = −∥x∥e_1$ Is $w = x − ∥x∥e_1$ not equal to ...
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1answer
13 views

Derivation of within point scatter $W(C)$

I'm reading the book "The Elements of statistical learning". In the section about K-means clustering they derive an equation regarding the "within point scatter" which is a ...
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0answers
30 views

Cauchy Schwarz manipulation

I am trying to manipulate the upper bound below using Cauchy Schwarz, so that the norm of the vector a, $||a||$, appears on the right hand side ($a_i$ itself is a scalar, $a$ is the vector). Which I ...
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1answer
32 views

Gradient of norms - general advice

I have something of the following sort: $$ F(x): \mathbb{R}^n \to \mathbb{R} $$ Where $F(x)$ is a function mapping from one value to another. For example, I may have functions of the form $$ F(x) = \|...
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0answers
19 views

Fenchel conjugate of $||.||_1$ and dual of logistic regression

I am trying to replicate some results from Koh, K., Kim, S. J., & Boyd, S. (2007). An interior-point method for large-scale l1-regularized logistic regression. Journal of Machine learning research,...
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1answer
25 views

norm my complex number to 1 [closed]

I have to find for a result that the product of a complex number and his conjugate is equal to 1. The problem is, my complex number is as sum of complex numbers If i take the simple case where it's a ...
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1answer
50 views

Gradient and Hessian of squared Frobenius norm

I want to find the Gradeint and Hessian of the following function, $F(\mathbf{S}) = \frac{1}{2}\Vert \mathbf{M} - \mathbf{K_2SK_1^T}\Vert _F^2+\frac{1}{2}\Vert\mathbf{S}\Vert_F^2$. My try: Using trace ...
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3answers
29 views

Addition/Subtraction of vectors regarding norm

If we subtract a vector with a certain norm from another one with the same norm what's the norm of the resulting vector? Since norm is just a length indicator it should remain the same, but I cannot ...
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1answer
21 views

Inequality on ratio of norms

I would have a question that is a specific case of this post - Bounding the ratio of the $l^1$ norms of two vectors to the ratio of their $l^2$ norms - with $x_j=y_j^2$ and $c_1=1$, $c_2=1/2$. ...
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1answer
18 views

Show operator norm attains value under Euclidean norm in $\mathbb{R^n}$

I've been trying to show that for $A \in M_{m\times n}(\mathbb{R})$, there is some $x \in \mathbb{R^n}$ with $||x|| = 1$ such that $||Ax|| = ||A||$, the operator norm of $A$. The definition of ...
3
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1answer
45 views

Sequence of functions that converges strongly in $L^2(\mathbb R)$ but not pointwise

I am trying to find a sequence of functions $\{f_j\}$ and a function $f$ such that $f_j\to f$ strongly in $L^2(\mathbb R)$ but $f_j$ does not converge to $f$ pointwise. The definition of strong ...
2
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2answers
36 views

Norm of convolution of $f$ and $g$ where $f \in L^1(R)$ and $g \in L^p(R)$

Here is the question: For $f \in L^1(R)$ and $g \in L^p(R)$, define $f*g(x)=\int_{- \infty}^\infty f(x-y)g(y)dy$ . prove that $f*g\in L^p(R)$ and $||f*g||_p\le||f||_1||g||_p$ This question is from ...
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2answers
30 views

Matrix norm proving calculation problem.

Okey here i am stuck in a problem in matrix norm proof, i don't need to the proof of Matrix norm , i just need to know how the calculations are done to part (1) and (2) in figure. Here is the complete ...
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1answer
46 views

Frobenius norm involving Kronecker Product

Consider $ J = ||\mathbf{G} - ( \mathbf{B} \otimes \mathbf{X} )||_F^2 $, where $\mathbf{G}$ and $\mathbf{B}$ are complex matrices, and $||.||_F$ is the Frobenius norm. Find the derivative with respect ...
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0answers
18 views

upper bound for quadratic form in terms of vector norm and eigenvalues

I have a quadratic form. if Q, P and M are positive and symmetric matrices. $$(-x^T Q x - 2 x^T Q e - e^T Q e) + (y^T M y + 2 x^T P y + 2 e^T P y )$$ how can I get an upper bound for this quadratic ...
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0answers
26 views

unique solution of l1 minimizer is sparse

I have tried to prove the following statement but after days of trying I couldn't! Suppose $\mathbf{X} \in \mathbb{R} ^ {N\times d}, \mathbf{y} \in \mathbb{R}^N$ and $\alpha > 0$. Show that for an ...
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3answers
92 views

Finding $\mathbf{x}$ that minimizes $\|\mathbf{x}-\mathbf{x_0}\|^2$ subject to minimizing $\|A\mathbf{x}-\mathbf{b}\|^2$

What is a good way to find $\mathbf{x}\in\mathbb{R}^n$ that minimizes $\|A\mathbf{x}-\mathbf{b}\|^2$ and subject to the above, minimizes $\|\mathbf{x}-\mathbf{x_0}\|^2$ where $A\in\mathbb{R}^{m\...
2
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1answer
29 views

Need an Upper Bound for $L^2$-Norm of Integral of a Gauss Function in 2 Dimensions

Statement of the Problem We wish to show that the following norm: $ \large || \int^{t/2}_{0} \xi_1 e^{-(t-s)|\xi|^{\alpha}} \int_{\eta \in \mathbb{R}^2} \frac{|\xi|^2 \eta_2 e^{-(S+1)|\xi - \eta|^{\...
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2answers
25 views

How to prove Cauchy Schwartz Inequality for norms in Lebesgue Integration

I am self studying Apostol ( Mathematical Analysis) but I couldn't prove this particular theorem given in text despite the hint given . So, I am asking here. Its part (e) , I have no idea how to use ...
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3answers
15 views

Quadratic matrix bounds

Let A be a singular matrix with a simple (non-repeated) zero-eigenvalue. Dose the following inequality hold? $$\|Ax\|^2\geq\sigma_2\|x\|^2, \qquad \forall x\notin Null(A)$$ where $\sigma_2$ is the ...
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1answer
43 views

Cauchy-Schwarz applied multiple times with difficulty on second application

Below, $a_i$ is a column vector with matching dimension as the square matrix $G$ and $b_{ij}$ is a scalar. I apply CS once over $(i,j)$ to obtain the first inequality but I'd like to apply CS once ...
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0answers
16 views

$C[a,b]$ is complete with the sup norm, proof verification

I know that this question has been asked before, but I want to share my thoughts on it, and ask for a proof verification: Let $C[a,b]$ be the vector space of every continuous function $f:[a,b]\to \...
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1answer
38 views

Counterexample for $\Vert A \Vert _{op} \cdot \Vert A^{-1}\Vert _{op} = 1$

$A \in GL_2$, invertible from $\mathbb {R}^2 \to \mathbb{R}^2$. I know that $\Vert A \Vert _{op} \cdot \Vert A^{-1}\Vert _{op} = 1$ isn't true. Can someone give me a counterexample?
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1answer
15 views

If $A \in L(\mathbb {R}^n)$ with $\Vert A \Vert _{{op}} \neq 0$, then $A \in GL_n$.

Which of these following statements is true? $n \in \mathbb {N}$ a) It exists $A, B \in L(\mathbb {R}^n)$ with $\Vert A\Vert _{{op}} \neq 0$, $\Vert B\Vert _{{op}} \neq 0$, but $\Vert AB\Vert _{{op}} =...
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2answers
40 views

Definition of the operator norm

I know that this definition is correct, is the bottom one also fine? $A \in L(\mathbb {R}^m,\mathbb {R}^n)$ $\Vert A \Vert _{op}:=\sup \{{\vert Ax\vert } \big \vert \,x \in \mathbb {R}^m,\vert \,\...
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1answer
30 views

Properties of ${\wr \hspace {-3pt}\wr \hspace {-1pt}A\hspace {-1pt}\wr \hspace {-3pt}\wr } := \inf \{\vert Ax\vert \,\big \vert \, \vert x\vert = 1\}$

For A $\in L(\mathbb {R}^n)$ define ${\wr \hspace {-3pt}\wr \hspace {-1pt}A\hspace {-1pt}\wr \hspace {-3pt}\wr } := \inf \{\vert Ax\vert \,\big \vert \, \vert x\vert = 1\}$. Which of the following ...
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1answer
21 views

k-mean clustering minimize L1 distance

k-mean clustering minimize L1 norm In k-mean clustering, if I want to minimize the L1 distance from any point to cluster center, the error function and derivative is shown above. However, according to ...
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1answer
22 views

Extension of median minimizing the sum of absolute deviations (the $L_1$ norm)

This is an extension of the question asked in The Median Minimizes the Sum of Absolute Deviations (The $ {L}_{1} $ Norm) . Except with the extra constraint that $x \in S$. The solutions provided there ...
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2answers
29 views

Folland Real Analysis Exercise 5.15 - A unique linear tranformation $S$ such that $T = S \circ \pi$ where $\pi$ is the projection onto $X/N(T)$

Consider the following problem, Exercise 15 of chapter 5 of Folland's Real Analysis, 2nd edition: Suppose that $X$ and $Y$ are normed vector spaces and $T \in L(X, Y)$. Let $N(T) = \{x \in X \ : \ Tx ...
1
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1answer
26 views

proves and disproves about inner product spaces

$l_p=\{[{a_n}]_{n=1}^ {\infty}|\sum_{n=1}^{\infty}|a_n|^p < \infty \}$ with the norm $||a_n||_p = (\sum_{n=1}^{\infty}|a_n|^p)^\frac{1}{p} $ prove or disprove: $L_2\subset L_1$ I know its true ...
1
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1answer
25 views

Norm of Position Operator in $L^2[0,1]$

I was wondering what is the norm of the position operator $Xf(x)=xf(x)$ in $L^2[0,1]$. I have two different results. The first one is the simplest and the reasonable: $$||X|| \overset{||f(x)||=1}{=} ...
0
votes
1answer
52 views

Prove that Fourier Coefficients given by the orthonormal projection the closest element to v in W

Let $V$ be an inner product space of finite dimension. Let $v_1, v_2, ... , v_m$ be orthonormal vectors in $V$ and $W=\operatorname{sp}\{v_1, v_2, ... , v_m\}$. let $v$ be some vector and $\alpha _i=\...
1
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1answer
17 views

norm and projections on inner product space

How do I show that if $\Vert Px-Qx \Vert <\Vert x \Vert$ for any $x\in V$ not $0$, then $\dim\left(M\right)=\dim\left(N\right)$. $V$ is an inner product space and $M, N$ are sub-spaces of $V$.$P$ ...
1
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1answer
24 views

Total variation norm of probability measures related to $L_1$-norm?

On Wikipedia the following is stated: I don't see how, if the set is countable, $$ \delta (P, Q) = \sup_{A \in \mathcal{F}} \vert P(A) - Q(A)\vert = \frac{1}{2} \vert \vert P- Q\vert \vert _1$$ holds....

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