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

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Trace class norm and rank inequality

I am quite new to operators in Hilbert spaces and I have been trying to show that for any linear and bounded operator $T : \mathcal{H} \rightarrow \mathcal{H}$ \begin{equation} \vert \vert T \vert \...
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lipschitz continuity on matrix product

If $ f(t,x)=A(t)g(t,x)B(t) $ where $ A(t), g(t,x), B(t)$ represents square matrix functions. If $ A(t), B(t)$ are bounded and $ g(t,x)$ is Lipschitz continuous. Then is it correct to consider $ |f(t,x)...
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Polar set of orthogonal matrices set is nuclear norm ball

Reltated problems: Show that the dual norm of spectral norm is Nuclear norm. Proof that nuclear norm is convex. The set of orthogonal matrices is defined as: $$\mathcal{O}(n) = \{X\in \...
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22 views

Relationship between matrix norms

Working in real space. Is spectral norm of a symmetric positive definite matrix greater than or equal to operator norm? Can you provide some inequalities between other norms like schatten norm, ...
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34 views

What is the difference between the infinity norm of a transfer function and the infinity norm of a matrix

I am studying robust control system, and get confused with the following two definitions of infinity norm. (G(jw) is the transfer function of a MIMO system) [1] $$\left \| G \right \| _\infty = \max \...
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Operator norm and eigenvalue inequality

Can I say that $\|A\| < s$ where $A \in \mathbb{R}^{3 \times 3}$ is a symmetric, positive definite matrix and $s$ is the maximum eigenvalue of $A$. Here the norm used is operator norm.
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Is $\mu A+\lambda B$ closed? [duplicate]

Let $E$ be a real normed space, $\mu>0, \lambda >0$ and $A,B \subset E$ closed and convex sets such that $0 \in A$ and $0\in B$. Is $\mu A+\lambda B$ closed? This is what I did so far: ...
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44 views

Show that the mapping $(C^1, ||.||_{C^1}) \rightarrow (C, ||.||_\infty) $ is continuous

Assume $D:(C^1, ||.||_{C^1}) \rightarrow (C, ||.||_{\infty}),$ $$D(f) = f',$$ is a mapping with $$||f||_{C^1} := ||f||_{\infty} + ||f'||_{\infty},$$ $$||f||_{\infty} := sup_{x \in [a, b]} f(x).$...
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1answer
16 views

Trace form of Frobenius Norm of Matrix approximation

I'm a CS Student and I've implemented the Convex Non-Negative Matrix Factorization (Convex-NMF) Algorithm for a project. Now, for "classic" NMF algorithms, you get an approximation: $$ \mathbf{A} \...
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27 views

Robusness of median

If we let $X$ be a set of pints in $\mathrm{R}^2$, and let $g(X) = \arg \min_{y \in \mathrm{R}^2} \sum_{x_i \in X} \parallel x_i -y \parallel_2$ (geometric median of $X$). If $X$ and $X'$ are ...
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Frobenius Norm of Projected Matrix

Let $A$ be a matrix of dimension $K \times N$ with complex entries. If $P$ is an idempotent projection matrix such that $P^H = P$, can we say that $Tr (APA^H)^{-1} \ge Tr(A A^H)^{-1}$? Note that the ...
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Proof of matrixnorm relative to one-norm

Let $A \in \mathbb{R}^{d \times d}$ a $d \times d$-matrix $A=(a_{ij})$ with norm $\|\cdot\|_1$. Proof: $$\|A\|= \max_\limits{j=1,...,d} \sum_\limits{i=1}^d |a_{ij}|$$ Let $\|x\|_1=1$ and $Ax=y$: $\|...
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31 views

Clarification of ideas concerning a quotient space.

Suppose I have a vector space $V$, and I identify $x\in V$ with $\lambda x\in V$, where $x\neq 0$ and $\lambda>0$, $\lambda\in\mathbb{R}$. I'm confused about two things: (1) Can I define a norm on ...
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1answer
58 views

Prove that an arbitrary norm is continuous. Is my proof correct?

Let $f: \mathbb{F}^n\rightarrow \mathbb{R}$ be defined by $f(a_1,\cdots, a_n)=\|\sum a_jv_j\|$. Show $f$ is continuous on $\mathbb{F}^n$. 1. $\|\cdot\|$ is an arbitrary norm on $\mathcal{V}$. ...
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Show that usual $L^2$ norm is equivalent to arbitrary norm $||\cdot||$ that satisfies 'convergence condition'

Given $L^2(\mathbb{R})$ consider a norm $||\cdot||$ on $L^2$ such that $(L^2,||\cdot||)$ is a Banach space and every $||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ...
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Showing this a norm

I want to show that $$\| x \| = \sum_{n=1}^{\infty} \frac{1}{2^n} \frac{\left| x_n \right|}{1+\left| x_n \right|}$$ is a norm. I'm fine showing positivity and the triangle inequality, to show the ...
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30 views

Inequality between norm of function and it's derivative

There is a theorem: Let $f$ be a continuously differentiable, $2\pi$-periodic function. Given $\int_{-\pi}^{\pi} f(x) dx = 0$, I need to prove that $$||f|| \le \frac{\pi}{2} \cdot ||f'||.$$ Where ...
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1answer
37 views

differentiability of the norm of L^1

Let $N_1$ denote the natural norm of the functional space $L^1(\Omega)$, where $\Omega$ is an open domain of $R^n$: $$ N(y)=\int_\Omega |y(x)| dx $$ I have the following question regarding $N_1$: ...
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65 views

Is there a generalisation of norm catering for $\|a\mathbf{v}\|=\|\mathbf{v}\|$?

I'm working with a function $p$ which gives a kind of "size" of the vectors in my vector space, and it has all the properties of a norm except that $$p(a\mathbf{v})=p(\mathbf{v}).$$ Ordinarily a norm ...
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2answers
41 views

Compact operator and norm

Let $E,F$ and $G$ be normed spaces, $f\in \mathcal{L}(E,F)$ and $g\in \mathcal{L}(F,G)$. Suppose that $g$ is injective and $f$ is compact. Show that, $\forall \varepsilon > 0$, there exists $M&...
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29 views

Which of following inequality holds?

$f(x,y) = _{\theta,\phi}^{{sup}}$ $ ||e^{i\theta}x - e^{i\phi}y||_{2}^2$ ,where $x,y \in \mathbb{C^2}$ and $\theta,\phi \in \mathbb{R}$ $f(x,y) \leq ||x||^2 + ||y||^2 - 2Re (\langle x,y \rangle )$ ...
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56 views

Triangle Inequality for $\|x\|_{\infty}$

I have to show the triangle inequality for $\|x\|_{\infty}$. I'm not sure, if estimate is correct. To show: $\|x+y\|_{\infty} \le \|x\|_{\infty}+\|y\|_{\infty}$ Let $x \in \mathbb{R}^n$ and $\|x\|_{\...
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21 views

Which of the inequality holds?

$f(x,y) = _{\theta,\phi}^{{sup}}$ $ ||e^{i\theta}x - e^{i\phi} y||_{2}$ ,where $x,y \in \mathbb{C^2}$ and $\theta,\phi \in \mathbb{R}$ Which of the following holds ? $f(x,y) \leq ||x||^2 + ||y||^2 ...
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1answer
27 views

Are the norms on a vector space unique?

I was watching an online lecture on bounded linear transformations $$T: \mathcal{C}[a,b] \rightarrow \mathcal{C}[a,b]$$ So the condition for $T$ to be bounded was that for all $f \in \mathcal{C}[a,b]...
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1answer
20 views

Vectors in Normed Space Must Have Finite Length?

I have assumed this to be the case, and consequently this is why one looks at convergent sequences of vectors in normed, Banach, and Hilbert spaces. But, I've never seen this listed explicitly as an ...
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Use operator norm to rigorously prove exp(ln(I + A)) = I + A

Show that $\exp(\ln(I + A)) = I + A$ when the operator norm of $A$ is less than 1. A similar question has been posted, Does $\exp(\ln(I+A))=I+A$ when $\|A\|<1$?, but this does not offer a real ...
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A Matrix Norm Inequality $\|A^{1/2}B^{1/2}(A+B)^{-1/2}\|_F \geq \|A^{1/2}(A+B)^{-1/2}B^{1/2}\|_F$

Let $\|X\|_F:= \sqrt{\text{Tr} \left(XX^\dagger\right) }$ denote the Frobenius norm. Does anyone know how to show the norm inequality: $\left\|A^{\frac12}B^{\frac12}(A+B)^{-\frac12}\right\|_F \geq ...
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How to bound difference of convex optimization problems, when the closed-form solution doesn't exit?

Let $z = \langle z_1 ,...,z_m \rangle$ where $z_i \in \mathbb{R}^d$, we define $g(z):= \arg \min_{y \in \mathbb{R}^{d}} \sum_{i=1}^{m} \parallel z_i -y \parallel_2$. We say $z \Delta z'=1$, if $z'$ ...
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32 views

Why do we study specifically 'normed' vector spaces?

When we study vector spaces, it is useful to define a norm on it for countless reasons. I was thinking about this recently and realised Don't all vector spaces have norms on them? If they all ...
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1answer
30 views

Norm of a Dirac's functional

Let be $\delta:(\mathcal{C}[-1,1], \| \quad \|_{sup})\to(\mathbb{R}, \lvert \quad \rvert)$ defined as $\delta:=2\delta_{-1}-3\delta_0+\delta_1$ with the Dirac functional $\delta_c\in\mathcal{C}[-1,1]'$...
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19 views

If possible calculate the norms of the following linear forms.

Let be $I$ and $d$ defined as $$I:f\to\int_{-1}^{1} f(x)dx \\ d:f\to f(0)$$ on $\mathcal{C}^1[-1,1]$. Check if the linear forms are continous with respektive to the norms $\|\text{ }\|_{\infty}, \|\...
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Loosely embedding $\ell_\infty$ in $\ell_2$.

Fix $n$ and $t$. I want a constant $C$ and a function $f : \mathbb{R}^n \to \mathbb{R}^m$ with the following properties. ($m$ can be arbitrary.) If $\|x-y\|_\infty \leq 1$, then $\|f(x)-f(y)\|_2 \...
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Norm of matrix and linear mapping with Riesz Representation Theorem

On page 175 of Edward's Advanced Calculus of Several Variables, there is the following theorem: Theorem 2.3 Let $A = (a_{ij})$ be the matrix of the linear mapping $L:\mathbb{R^n} \rightarrow \mathbb{...
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Showing $\|x+y\|^2+\|x-y\|^2=2\|x\|^2+2\|y\|^2 \Rightarrow \| \cdot \| $ is induced by scalar product

I need to show the above $\forall x,y,v \in V$ , a normed vector space on $\Bbb R$. A hint was given that i should first show that $$s:V \times V \to \Bbb R ; \: \:\: s(u,v):=\frac1 4 (\|u+v\|^2-\|u-v\...
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51 views

Show that $\ker(T)=\{\varphi _n\mid\lambda_n\neq 0\}^\perp $

Let $T:H \to H$ be defined as $Tx=\sum_{n=1}^{\infty} \lambda_n \langle x,\varphi _n \rangle \varphi _n$, given that $\{\varphi _n\}_{n=1}^\infty$ is an orthonormal sequence (not necessarily a basis) ...
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28 views

Show that $\|F\|=sup_n |\lambda _n|$.

Let $F:H \to H$ be defined as $Fx=\sum_{n=1}^{\infty} \lambda_n \langle x,\varphi _n \rangle \varphi _n$. given $\{\varphi _n\}_{n=1}^\infty$ is an orthonormal sequence (not necessarily a basis) and $...
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Which norm is applied in Gram–Schmidt process?

I am following the wiki entery for Gram–Schmidt process and it states that: $e_1=\frac{u_1}{\|u_1\|}$ I can see they are using the $\|\cdot\|_2$ norm in their example, But if I am applying this ...
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On an inequality concerning vectors with norm less than one

Consider two sets of vectors $\{v_i\}_{i=1}^{n_1}$ and $\{w_j\}_{j=1}^{n_2}$ with $v_i,w_j\in\mathbb{R}^n$ such that $\|v_i\|_2< 1$ and $\|w_j\|_2< 1$ for all $i=1,\dots,n_1$ and $j=1,\dots,n_2$....
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Are there any results concerning norm/spectral radius product equalities?

For square matrices $\mathbf{A}$, $\mathbf{B}$, some norms (I am dealing with the spectral norm) satisfy $\| \mathbf{AB} \| \leq \|\mathbf{A}\| \|\mathbf{B}\|$. Can we say anything about the case ...
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1answer
46 views

Compute H-infinity norm in Matlab

Please can someone write a command in Matlab for calculating $H_{\infty}$ norm for the following system: $$\frac{d}{dt}z(t)=Az(t)+Bu(t)+Fw(t)$$ $$y(t)=Cz(t)+Du(t)$$ where $A$, $B$, $C$, $D$, and $F$ ...
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44 views

Inequality of Scalar Product involving derivative

I am stuck trying to reach an (in)equality... Let $\Omega \in \mathbb{R}$ and $f=f(t,x): \mathbb{R} \supset[0,T] \times \Omega \rightarrow \mathbb{R}$ be an element of $$\mathcal{W}:=L_{_t}^2(0,T,H_{...
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Entrywise expression for L2 matrix norm

The matrix norm induced by the $\ell^2$ norm is known to be equal to the maximum singular value of the matrix. The matrix norms induced by the $\ell^1$ and $\ell^\infty$ norms admit simple ...
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completion of $C_{00}$ with different (semi)norms

Let $C_{00}(I)$ be space of complex sequences indexed by countable set $I$ with entries are zero for all but finitely many entries. We endow $C_{00}(I)$ with the following family of (semi)-norms: $\...
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Dual norm to a matrix norm corresponds to the matrix norm of the inverse [duplicate]

Let $A$ be a positive definite matrix. Then one can define the matrix norm $$\|x\|_A := \sqrt {x^T A x}.$$ The dual of a norm is defined as $$\|x\|_A^* := \max_{\|y\|_A\leq1} \langle y,x\rangle.$$ It ...
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Computing unknown matrix norm

Suppose that we have a unknown vector norm but there is a machine that get a vector and correctly tell us the norm of the vector. Is there any algorithm to compute the matrix norm corresponding to ...
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39 views

Norm of an operator in a Hilbert space

Let $T\neq 0, \neq I$ be a linear operator of a Hilbert space such that $T \circ T = T $. Show that $\|T\|=\|I-T\|$. Anyone has an idea ? I just proved that $\|I-T\| \leq 1 + \|T\|$ but it is not ...
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36 views

prove $||A_n^{-1}||$ is bounded

Let $(A_n)_{n=1}^\infty{} \in GL_{n\times n}(\mathbb{R}) $. $\lim_{n\rightarrow\infty} A_n = A$; $A\neq0$ is invertible. I have a notion that for any norm $$||\cdot||:GL_{n\times n}(\mathbb{R}) \...
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Intuitive explanation of $L^2$-norm

I have to play a lot with the $L^2$-norm defined as $\|w\|=\sqrt{\int_a^b <f,f>}$. However, I don't understand the interpretation of that norm. We know that the euclidean norm measure the length ...
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27 views

Why the ideal norm is multiplicative

Let $I\subseteq B$ be an ideal, we define the ideal norm of $I$ as the ideal in $A$ generated by the elements $N_{E/K}(\alpha)$ where $\alpha \in I.$ We denote it by $N_{E/K}(I).$ If $\mathfrak{p}$ ...
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70 views

What are vector norms used for?

I'm currently working with a computer science problem that requires me to build vectors that can return their own norms. Based on Wolfram Alpha's description, I think I have an idea of how this is ...