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

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An Orthogonal Projection with Weighted Norm

In the context of solving a convex program via projected gradient descent i am facing the following problem: $$\min_{x\in\mathbb R^2}\lVert x-y\rVert_M^2,\qquad\lVert x\rVert\le1$$ or written ...
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Seminorm proof of a function

I have an example in a book which is not very clear to me : let $E$ vector space made of numerical functions (or complex) $f$ defined on a set $A$. $\forall a \in A, N_a : f \rightarrow |f(a)|$ is a ...
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48 views

Are isometries always linear?

Let $E$ be a finite dimensional vector space (over a field of characteristic zero) and $f : E \rightarrow E$ be an isometry fixing 0. Must $f$ be linear in this case ? Note : I am NOT assuming that ...
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Bounding Sobolev Norms *Below*

Firstly, apologies if this is a duplicate question - I've looked, but can't find this question on SE (or elsewhere online); if it is, please let me know and I'll remove it. I am trying to bound the ...
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54 views

Norm of a Vector

Suppose $A\inℝ^{n,n}$. We Define $$ \|A\| = \underset{\|x\| = 1}{\sup} \frac {\|Ax\|} {\|x\|}$$ Show it is a norm. Any thoughts?
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Well-Posedness PDE of the Form $\partial_t u = P(\partial_x) u$ for a Polynomial $P$

My question is to determine whether the PDE $\partial_t u = P(\partial_x) u$, with $2\pi$-periodic boundary conditions, for a polynomial $P$, is well-posed; this depends on the polynomial, and my ...
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Evaluating the distance beween two vectors of different dimensions.

I am asking for a way (if possible) to evaluate the distance between two vectors $p=(p_1,....,p_{n})∈ℤ^{n}$ and $q=(q_1,....,q_{m})∈ℤ^{m}$, where $n \neq m$.
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Is a norm on $R^n$ linear?

I was reading the book Linear Algebra Done Right by Axler. In the chapter on inner product space (Ch.6), he defines the norm of x on $R^n$ space as: $||x|| = \sqrt{x_1^2 + ... + x_n^2}$ and says: ...
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38 views

Understanding a statement about equivalent norms ($||\cdot ||_2 \sim||\cdot||_1 $)

I am trying to understand the following statement from an analysis book: Two norms are equivalent ($||\cdot ||_2 \sim||\cdot||_1 $) if they induce equivalent metrics. At first I thought this ...
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20 views

Minimum in complex inner product vector space

I'm stuck at this problem, can someone give me a hint? Let $x_i$ and $y_i$ ($i=\overline{1,n}$) be vectors in an infinite dimensional vector space $V$ with inner product $(,)$ satisfy: ...
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54 views

Norm of symmetric positive semidefinite matrices

I have been researching a lot trying to find an answer to my question and didn't find any so I would appreciate it if anyone can help. If we have 2 symmetric, positive semi-definite matrices $A$ and ...
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36 views

Where does the definition of the $L_0$ norm come from?

Where does the definition of the $L_0$ norm come from? $$\|x\|_0=|S|$$ Where $S=\{x_k:x_k\neq 0\}$
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How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?

I have two segments in $\mathbb{R}^2$ and I would like to define a measure of similarity between the two segments. My idea is that I can apply: a scale transformation $s$ in order to equate the ...
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55 views

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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Equivalence of sets

Let $u_1, u_2, u_3 \in \mathbb{C}$ be the cubic roots of unity I'm wondering if the following two sets (balls) are equivalent: $$ \lbrace (v,w) \in \Bbb C^2 : \vert v \vert + \vert w \vert \leq 1 ...
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31 views

Is the norm of the average $\le$ the norm of the max?

Given $\pmb X \in \mathcal{R}^p$, denote the elements of $\pmb X$ as $\pmb x_i$ for $i= 1, \dots, n$. Denote the $t(\pmb X)$ as the average of $\pmb X$ \begin{equation} \pmb t(\pmb X) = \frac 1 n ...
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19 views

Is $||A||_F ||x||_2^2 \geq x^TAx$

Given a symmetric matrix $A$ and a vector $x$ Is $||A||_F ||x||_2^2 \geq x^TAx$? If yes, how to show this?
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94 views

On an estimate of sequences with weights

Does there exist a $C > 0$ such that $$ \sum_{n \geq 1} a_n \leq C \left( \sum_{n \geq 1} 2^n a_n^2 \right)^{1/4} \left( \sum_{n \geq 1} 2^{-n} a_n^2 \right)^{1/4} $$ for all $a_n \geq 0$ with ...
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1answer
33 views

Why is $L_0$ norm not convex? [closed]

I have this confusion in understanding the convexity of the $L_0$ norm. Why is $L_0$ norm not convex?
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21 views

Geometric characterization of an Euclidean norm

Show that $N$ is an Euclidean norm if and only if the intersection of the unit ball with any plane is an ellipse. I'm stuck on this one. I do not see how can I connect the definition of an ...
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29 views

Express Norm Using Inner Product

I'd like to know whether there's a way to express a norm using inner product, for example , is there any inner product we may use that is equal to $(||Ax-b||_2)^2$ ? Thanks in advance.
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$L^1$ norm of convolution

Let $f_{\lambda} = \frac{\lambda}{2}e^{-\lambda |x|}$. Prove that $||f_{\lambda} \ast g - g || \to 0$, when $\lambda \to \infty$, where $g \in L^1$
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Matrix one-norm and infinity-norm

Help me please to find $3\times 3$ matrices $A$ and $B$ under following conditions: $\left \| A \right \|_{\infty }=4\left \| A \right \|_{1}, A \neq 0$ $\left \| B \right \|_{1}=4\left \| B \right ...
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Convexity of Norm of Max

Let $p \ge 1$. Show the convexity of the function $h:\mathbb{R}^k \rightarrow \mathbb{R}$ defined as: $$h(\textbf{z})=\left(\sum\limits_{i=1}^k \max\{z_i,0\}^p \right)^{1/p}$$
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Why is $\Vert xy\Vert>\Vert x\Vert \Vert y\Vert$ not allowed?

If we look at the usual norm on $\mathbb R$, i.e. $\vert\cdot\vert$, then we see that $\vert xy\vert=\vert x\vert\vert y\vert$. Untill now I've just assumed that this propperty also holds for norms in ...
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36 views

Value of $\frac1{\Vert f\Vert}$

For $V=\{x\in X\mid f(x)=1\}$ show that $\inf\{\Vert x\Vert\mid x\in V\}=\frac1{\Vert f\Vert}$, where $X$ is banach and $f$ is a nontrivial element of the dual space of $X$. For $x\in V$ we have ...
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48 views

Why is L21 norm not smooth

I have this confusion. I was reading this paper http://www.cis.temple.edu/~yuhong/research/papers/ijcai13b.pdf. I didn't understand why is L21 norm not smooth?
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28 views

Is taking the Euclidean norm of multiple Euclidean norms equivalent to taking the Frobenius norm?

I'm just a programmer venturing into the world of norms (is that even a thing?) here, and am wondering if two formulas are equivalent. Please forgive my ignorance! Suppose we have a $10\times3$ ...
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Bounding Vector Norm

Let $\theta, \mu$ be vectors in $\mathbb{R}^n$. And suppose we have the relation, $$ \theta = \arg \max_{\theta'\in\mathbb{R}^n} \left\{\left(\theta'\right)^T \mu - A\left(\theta'\right)\right\} $$ ...
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$\frac{\|\mathcal Ax\|_{n}}{\|x\|_n}$ 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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14 views

Matrix norm can assume infinite values?

Given a real-valued matrix $A=a_{i,j} \in M_{n,n}$ When: $||A||_2= \sqrt {\sum_{i=0}^n a^2_{i,j}} < + \infty$ ? Why?
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Space of Distribution wrt to topology of uniformly convergence on bounded sets not Frechet-Space.

I found a state, that the Space of Distribution on (here:) $\mathbb{R}^{n}$, which is equipped with the topology of uniformly convergence on bounded subsets is not a Frechet Space. As far as i can ...
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Prove or disprove the existence of a length preserving non-normal matrix

Prove or disprove: There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a normal matrix There exists A such that $\|Av\|=\|v\|$ for any $v \in R^n$, but $A$ is not a unitary ...
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$\exp(\ln(x))=x$ 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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Bounding the norm of Gaussian random matrix

Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral ...
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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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Operator in $\mathbb R^2$

I am a bit confused, can someone help me with the following? Is there an operator $T$ in $\mathbb{R}^2$ such that: $\parallel u \parallel +\parallel v\parallel = \parallel T(u+v)\parallel$ for every ...
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37 views

Show that a set is compact.

Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for ...
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38 views

Find the norm of functional

Consider the functional from $l_2$. $$ x=(x_n)\mapsto \sum \frac{x_n+x_{n+1}}{2^n}. $$ What is the norm of the functional?
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32 views

How can you compare two mathematical functions of single variable? [closed]

I have some functions and want to see how close one function is to the other. One way to check is correlation. What are the other ways of checking the "similarness", "closeness" or "nearness"?
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Show that $\int_{\mathbb R^n}e^{|x|^{-n}}dx=$ Volume of n-sphere

I'm preparing for a calculus exam, I'd like help in solving this question. Let $x \in \mathbb R^n$, $|x|={(x_1^2+x_2^2+...+x_n^2)^{\frac{1}{n}}}$, Show that $$\int_{\mathbb R^n} e^{|x|^{-n}}dx$$ is ...
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Lengths of curves - Arc length

If $\gamma\colon [a,b] \rightarrow \mathbb{R}^d$ is a differentiable curve we usually define $$ \text{length}(\gamma) := \int_a^b \|\dot{\gamma}(t)\|_2\, \text{d}t,$$ where $\|\cdot\|_2$ is the ...
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Find norm of linear operators

I have to check if those operators are bounded and if so what are their norms. 1) $\phi:C^1[0,1]\ni f > \rightarrow\int_0^{1/2}f(t)dt+f'(2/3)\in\mathbb{R}$ with norm ...
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Relationship between matrix 2-norm and orthogonal basis of eigenvectors

Given the following matrix: $$ A = \left( \begin{array}{cc} 3 & 4 \\ 0 & 5 \\ \end{array} \right)$$ calculate $\|A\|_2$, with $\|A\|_2 = max_{x \in \mathbb{R}^2 -\{0\}} \frac{\langle Ax,Ax ...
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Norm and InnerProduct Inequality

How can I show that this is true: Let $u,v \in \mathbb{R}^n$: \begin{align} \frac{\|u\|}{\|v\|} \leq \frac{(u,u-v)}{(v,u-v)}, \quad \hbox{if} \quad (v,u-v) > 0 \end{align} Where $\|\cdot\|$ is ...
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19 views

Open set generated by two equivalent norm

Suppose X is an vector space and $||.||_a$ and $||.||_b$ are two norm on it, how i can proof that ''This two norms are equivalent iff they generate same open sets."? P.S.: Sense of question made by ...
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Projection and matrix norm

Suppose we are in the matrix space $\mathbf{R}^{n_1 \times n_2}$. Suppose, $R_{\Omega}$ is an operator, such that $R_{\Omega}(Z)$ chooses $m$ entries from $Z$ uniformly at random with replacement and ...
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31 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 $~$ ...
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16 views

What to call the Euclidian norm divided by a constant

I'm using the Euclidian distance $d_{2}$ divided by a constant $T$, i. e. $\frac{d_{2}}{T}$. However, I'm not sure what to call this. I'd like to keep things simple so I thought maybe "scaled ...
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Matrix Norm Inequalities and Linear Least Squares

I am working through one of my universities old QUALS and came across the following problem: Let $A,E\in \mathbb{C}^{m\times m}$. Suppose that $\sigma_\min>0$ is the smallest singular value of ...