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

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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: ...
3
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1answer
40 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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1answer
35 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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68 views

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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4answers
53 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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1answer
21 views

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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1answer
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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1answer
18 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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1answer
90 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
30 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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1answer
19 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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32 views

$L^1$ 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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1answer
26 views

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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65 views

Norm inequality? [on hold]

What are best possible constants $K_1,K_2,K_3$ would the inequality \begin{align} K_1||f||_2||g||_2 \le K_2||f||_p||g||_q \le K_3||f||_1||g||_{\infty} \end{align} holds? for all $p>1$ with ...
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1answer
35 views

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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2answers
138 views

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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1answer
35 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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1answer
46 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?
3
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1answer
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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13 views

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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4answers
58 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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1answer
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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1answer
42 views

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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1answer
32 views

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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140 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 ...
2
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0answers
19 views

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 ...
3
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2answers
61 views

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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1answer
20 views

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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1answer
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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1answer
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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1answer
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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1answer
51 views

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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1answer
31 views

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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44 views

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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1answer
25 views

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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2answers
32 views

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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1answer
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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23 views

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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1answer
28 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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1answer
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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1answer
57 views

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 ...
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30 views

norm of a nilpotent matrix

A proof I was reading used the claim that $||{N}||_2$ = 1 for a nilpotent matrix $N$. I tried to prove it, and have a couple of questions on it. First, my "proof": We know that there exists a basis ...
3
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1answer
92 views

Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but ...
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1answer
40 views

Inequality with a norm

I need help with the following: Let $A=\left(\begin{array}{cc}a & b \\c & d\end{array}\right)$, with $a\in\mathbb{R}$, $b\in(l^{1})^{*}$, $c\in l^{1}$, and $d\in L(l^{1},l^{1})$. Let $h\in ...
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1answer
71 views

Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$

Let $(x_n)$ be weakly convergent, but not norm convergent, sequence from a Banach space. Show that $\limsup_{n\rightarrow\infty} \|x_n\|^{\frac{1}{n}}=1$. Any help?
4
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72 views

A norm on $\mathbb{R}^2$ such that $\partial C$ is the unit sphere?

Suppose we are on $\mathbb{R}^2$. Assume that $C \subset \mathbb{R}^2$ is a convex bounded neighborhood of the origin invariant by central symmetry. Let $\partial C$ denote the boundary of $C$. My ...
2
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2answers
53 views

Condition for a norm be absolute

Let $\|x\|_B\mathrel{\mathop:}=\sqrt{x^{t}Bx}$, where $B \in \mathbb{R}^{n\times n}$ is a symmetric and positive semidefinite matrix. If $\mid x\mid = (|x_1|,|x_2|,\ldots,|x_n|)$, I want to show that ...
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1answer
53 views

$||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$

$||f + g||_{L^p} = ||f ||_{L^p} +||g||_{L^p} $ for $p\in\mathbb{R}^+\setminus\{ 1\}$ and $f,g\geq 0$, then $f = Cg$ for some non-negative constant $C$. First assume $||f ||_{L^p} +||g||_{L^p} = 1$, ...
2
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1answer
29 views

Proof that frobenius norm is a norm [duplicate]

It's pretty basic and I'm sure I'm missing something dumb here, but I'd like to know why $||A+B||_F \leq ||A||_F+||B||_F$ The way I understand it, ...