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

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Show that $\| *\|_p$ represents a norm on $C([0, 1]).$

A few weeks ago, I had to work on the following excercise: Assume, $$\|f\|_p := (\int_{0}^1 |f(x)|^p)^{1\over p}$$ with $f \in C([0,1])$ and $p \in [1, \infty)$. Show that $\| *\|_p$...
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31 views

What are all the Banach algebras where $\|a\|\|a^{-1}\|=1$? [on hold]

Is there a characterization for all Banach algebras such that $\|a\|\|a^{-1}\|=1$ whenever $a$ is invertible?
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38 views

A confusion about the definition of the “trace” norm

Given a $n \times m$ real matrix $A$ of rank $r$ one can define its SVD as $A = UD V^T$ with $D$ being a $r \times r$ diagonal matrix and $U^TU = V^TV= I$. Here clearly the diagonal entries of $D$ are ...
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16 views

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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159 views

What are some usual norms for matrices?

I am familiar with norms on vectors and functions, but do there exist norms for spaces of matrices i.e. $A$ some $n \times m$ matrix? If so, that would that imply matrices also form some sort of ...
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33 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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37 views

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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1answer
56 views

Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find square root for Number Field Sieve(reference here page 4 first line). It says that for this method the degree of extension K/Q must be odd so that Norm(-x)=-...
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8 views

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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67 views

What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
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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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31 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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1answer
38 views

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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15 views

Proving that a product norm does not come from a scalar product

We have nonzero normed spaces $X_1$, $X_2$. On the product $X_1 \times X_2$ we put the norm: $$ ||(x_1,x_2)||=||x_1||_1+||x_2||_2$$ and would like to show that it does not come from an inner product. ...
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14 views

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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43 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
11 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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1answer
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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25 views

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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27 views

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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23 views

Bound $L^{2}$ norm

I have an $L^{2}$ norm say $||v \ \partial_{xxx} u||^{2}_{L^{2}}$ and $u \in H^{3}$. Can I write $||v \ \partial_{xxx} u||^{2}_{L^{2}} \leq ||v||^{2}_{L^{2}} \ ||u||^{2}_{W^{3,\infty}}$
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56 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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31 views

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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1answer
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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1answer
507 views

Matrix inner product, and operator and trace norm inequality

I have trouble proving the following inequality. Let a matrix $A \in \mathbb{R}^{M \times N}$, and $\sigma_i(A)$ be the i-largest singular value of A. Define the operator norm and the trace norm as ...
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26 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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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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40 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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Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda x||=...
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1answer
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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90 views

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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29 views

Inequalities in weak $L^1$ norm

I have the following lemma: Suppose that for $j=1,2,\dots,$ $g_j(x)$ is a nonnegative function on a measure space for which $\left|\left\{x: g_j (x) >s\right\}\right|<1/s$. Let $\{c_j\}$ be a ...
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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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808 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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14 views

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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30 views

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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1answer
29 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
27 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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1answer
48 views

Convergence of linear functional

I want to determine wether the functional $\varphi_n:\ell^2\to \mathbb{R}$ defined by $$\varphi_n(x)=\frac{1}{n}\sum_{k=1}^n\sqrt k x_k\quad x=(x_1,x_2,\dots)$$ converges in norm, or in weak sense. ...
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20 views

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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23 views

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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49 views

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\...
2
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2answers
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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1answer
27 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 $...