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

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$|Av||A^{-1}v|$, $A$ non-singular, $|v|=1$.

Let $A$ a non-singular $n \times n$ matrix, $v \in \mathbb{R}^{n}$ a variable vector. The operator norm of $A$ is defined to be $|| A ||=\max_{|v|=1} |Av|$ where $|Av|$ is the standard Euclidean norm ...
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Frobenius Norm Unitary Operators

For something I'm working on, I have a matrix $A$ with other matrices $U$ and $V$ which are unitary ($U^*U = I$ and $V^*V = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\| ...
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45 views

Inequality of scalar-product and norm

Why does the following inequality hold, given $A$ is symmetric and $\lambda_{\min} (A)$ is the smallest Eigenvalue of $A$? $$v^\top A v \ge \lambda_{\min} (A) \; ||v||^2$$
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Following positive semi-definitness from matrix norm

How can I follow the following? $$||A||_2 \le \sigma > 0$$ $$\Leftrightarrow A -\sigma I \mbox{ is positive semi-definit}$$ I always get it the other way around, i.e. that $\sigma I - A$ is ...
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Frobenius Norm Triangle Inequality

How can I go about proving the triangle inequality holds for the Frobenius norm? I worked through $\|A+B\|_F \le \|A\|_F + \|B\|_F$ and was not able to make it work =/.
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Frobenius Norm Properties

Say $v\in {\mathbb{R}^{l}}$ (a column vector) and $A\in {\mathbb{R}^{l\times l}}$. I have encountered the following equality: $||v{{v}^{T}}-A||_{F}^{2}={{\left( {{v}^{T}}v ...
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44 views

Integral like a norm instead of sum

A Rienmann integral is defined as: $\int_a^b f(x)\ dx=\displaystyle\lim_{n\rightarrow\infty}\displaystyle\frac{1}{n}\displaystyle\sum_{i=0}^{n}f\left(\displaystyle\frac{b-a}{n}\cdot i+a\right)$ I ...
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Frobenius Norm with Unitary Operators

For something I'm working on, I have a matrix $A$ with another matrix $U$ which is unitary ($U^*U = I$), and I'm trying to show that, for the Frobenius norm, $\|A\| =\|UA\|$. Now, I can do this ...
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43 views

If $||f-g|| < ||f^{-1}||^{-1}$, then $f$ is isomorphism implies $g$ is also isomorphism

Let $E$, $F$ be Banach space, $f,g \in L(E,F)$ and $f$ is isomorphism. Prove that if $||f-g|| < ||f^{-1}||^{-1}$, then $g$ is isomorphism. Hi everybody. I got stuck on this problem and can't ...
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105 views

Plot of ||X||infinity norm

Can anybody tell me why the plot of $\|X\|_{\infty}$ in $\mathbb{R}^2$ comes out to be square? Since $\|(x_1,x_2)\|_{\infty} = \max\{|x_1|,|x_2|\}$, then let us say $|x_1|$ is max. Why the plot is ...
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53 views

A normed space of continuous functions with norm $\int_{0}^{1}|f(t)|dt$ is not complete

Suppose $E$ is a normed space of all continuous functions on $[0,1]$ with norm $\int_{0}^{1}|f(t)|dt$. Prove that $E$ is not complete I know that we must do is to find a Cauchy sequence of ...
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75 views

Can $||f|| = ||a||_{q}$ to arbitrary values of $p$ and $q$ satisfying ${1 \over p} + {1 \over q} = 1$

We all know that: Suppose $a = (a_{1}, a_{2}, ..., a_{n})$ is a point in Euclide space $R^{n}$. Consider the mapping $f: R^{n} \rightarrow R$, $f(x) = \sum_{i=1}^{n}a_{i}x_{i}$. Then $||f|| = ...
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82 views

Hölder Space Definition

At the beginiing of the defintion of Hölder spaces and the Hölder space norm. They start defining the first term of the Hölder norm as follows: If $u:U \rightarrow \mathbb{R}$ is bounded and ...
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336 views

Condition Number of a block Matrix

Is this hypothesis true? $$cond([A,B])≤cond(A)+cond(B)$$ where $cond$ is the Condition Number. And is this true for rectangular matrices($nxm$)? Let's consider $3$ different conditions for $A$ and ...
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56 views

Generalized minimal residuals: eigenvalues and sets of functions

Can someone help me on this exercise (2 parts)? Thanks! Suppose that $S \subseteq \mathbb{C}$ is a set whose convex hull contains $0$ in it's interior (so $S$ is contained in no half-plane ...
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125 views

Equality involving norm on Cholesky/QR factorization

Let $B$ be symmetric and positive definite and $B=B^{\frac{1}{2}}B^{\frac{1}{2}}$ the Cholesky factorization. Having $A=QR$, why can we follow the last equality in the following? $$ || A ...
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331 views

inequality with the Frobenius norm for matrices

Let $A\in M_n$. How can I show that $$\left|{\textrm{Tr}(A)\over\sqrt{n}}\right|\leq \Vert A\Vert_F$$ I tried it using the Cauchy-Schwarz inequality.
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Vector norm Inequality proof

Does anyone know how to start proving this inequality $$ \left\|\frac{x}{\|x\|} - \frac{y}{\|y\|}\right\| \leq \frac{4 \|x-y\|}{\|x\|+ \|y\|} $$ The norm is a random norm on a vector space $V$
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$\lambda_k \to 0$ implies $T$ is compact?

I am doing an exercise which asks to show that if $\{\varphi_k\}$ is an orthonormal basis in a Hilbert space with $T$ a bounded operator satisfying $T\varphi_k = \lambda_k \varphi_k$, then $\lambda_k ...
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50 views

Methods of computing the derivative of vector norms

I am very new to norms. Except the basic definitions and properties of the norm, I don't know too much about it. Now, I am very interested in computing the derivative of the norms. So, I am wondering ...
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200 views

Proof that two norms $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ are equivalent

Two norms $\def\norm#1{\lVert#1\rVert}\norm\cdot_1$ and $\norm\cdot_2$ are equivalent iff $\;\exists\;c_1,c_2>0$ such that $c_1\norm x_1\le \norm x_2\le c_2\norm x_1$ Show that $\norm ...
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continuity of norms with respect to $p$

Let $f\in L^{\infty}(\Omega,\Sigma,\mu)\cap L^{1}(\Omega,\Sigma,\mu)$. Then $w(p)=||f||_p$ is continuous function of $p$ for any $p\in [1,\infty)$. How to prove this? I have obtained the proof that ...
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238 views

Norm of random vector plus constant

Suppose that $w$ is a multivariate standard normal vector and $c$ a real vector of the same size. I know that for positive x $$P(||w+c||^2\geq x)\ \geq \ P(||w||^2\geq x)$$ but I cannot prove it. We ...
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64 views

Does this show the norm of this operator is zero?

We have $$T: C[-1,1]:\to \mathbb{R}$$ $$T(f)=\int_{-1}^1 x f(x) dx$$ The norm considered in $C[-1,1]$ is $$||f||=\max_{x\in[-1,1]} |f(x)|$$ So using $$||T||=\inf\{M:||Tf||\leq M||f||\}$$ in this ...
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272 views

does the max function holds the triangle inequality?

I need to prove if the following is a norm: $$||f||:=\max_{-1<x<0}|f| + \max_{0<x<1}|f|$$ when $f$ is a continuous on $[-1,1]$. The only problem I have is with showing it holds the ...
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145 views

Condition Number of a Product

Is this hypothesis true? $$\operatorname{cond}(AB) \leq \operatorname{cond}(A)\operatorname{cond}(B),$$ where $\operatorname {cond}$ is the condition number. And is this true for rectangular matrices? ...
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293 views

Using the Sobolev-Nirenberg-Gagliardo inequality in a proof

If $1 \leq p < n$. The Gagliardo-Nirenberg-Sobolev inequality states that there exists a constant C such that $||u||_{L^{p^{*}}(\mathbb{R}^{n})} \leq C||Du||_{L^{p}(\mathbb{R}^{n})}$ for all $u ...
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Triangle inequality for $\sqrt{x^T\cdot A\cdot x}$

I want to prove that $f(x):= \sqrt{x^TAx}$ is a norm, $A \in \mathbb{R}^{n \times n}$ positive definite, $x \in \mathbb{R}^n$. I already proved its positivity and absolute homogeneity, but I don't ...
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Limit of nth power of operator norm

I am given a compact operator $A$ which lives in a Banach algebra and whose spectral radius obeys $\rho(A)=\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}<1$. Now I want to prove that this implies ...
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60 views

Given identity map $U:\ell^n_a\rightarrow \ell^n_b$ for $\mathbb{R}^n$, how to computer operator norm $\forall a,b$?

If $U:\ell^n_a\rightarrow \ell^n_b$ is the identity map of the underlying vector space $\mathbb{R}^n$, then how do you compute the operator norm $U$ for all possible values of $a$ and $b$?
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56 views

Hints on calculating an integral

Suppose $x(t)$ and $y(t')$ are curves traversing the boundary of $[0,1]^2$ in $R^2$ counterclockwise. What is the integral of the following: $$\int\limits_{t,t'}{\frac{dt\,dt'}{\|x(t)-y(t')\|}}$$ I ...
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Showing that function are equal almost everywhere in Sobolev Spaces

Consider the Holder space $C^{0,1-\frac{n}{p}}(\mathbb{R}^{n})$ and the Sobolev Space $W^{1,p}(\mathbb{R}^{n})$. Take $u_{m} \in C_{c}^{\infty}(\mathbb{R}^{n})$ such that Morrey's Inequality we have ...
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Space on convergent sequences - is it an inner product space

I'm trying to prove some properties of sequence spaces. I already know that the space $l^{\infty}$ of all bounded sequences isn't an inner product space, isn't separable but it is complete with $sup$ ...
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44 views

If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every subordinate norm.

I need some guidance showing that: If a matrix of the form I+B is singular, where I is the identity matrix, then for any subordinate norm $\|\cdot\|$, $\|B\|\geq1$.
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27 views

Norm of the maximum

Consider the norm $||f||= max_{x\in[a,b]} |f(x)|$ defined in the bectorial space $C[a,b]$ I have to what is the meaning (/interpretation) in $R$ of {$||f_n-f||$}$\to$ 0 Could you help me?
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A norm for Lipschitz-continuous functions

I am a first year undergrad math student, and I am struggling with a proof. Let the set $C_{\text{Lip}}:= \left \{ f:\mathbb{R}\rightarrow \mathbb{R}: f \text{ is Lipschitz continuous} \right \}$ be a ...
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73 views

Minimization of norms

How do I minimize the following? $ min_{z>0} - zt + 1/2\ z\ ||\ Y + X_k\ /\ z\ ||_2^2 $ Also, $X_k^TX_k = 1 \ \ \forall k $ I am given that the answer should be : $ \sqrt{Y^T - 2t} + Y^TX$ ...
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37 views

Square of 2-norm

This might be silly but I am stuck with the following problem: $ || Y - Z_i/x||^2_2 $ = 2t How would I solve to get $x $ from this equation?
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What could be the lead to prove $||X||_2 \leq ||X||_F \leq \sqrt{rank(X)}||X||_2$?

In the above statement, $||X||_2$ = $L_2$ norm of X and $||X||_F$ = $Frobenius$ norm of X. It appears to me that the $L2$ norm of X and $Frobenius$ norm of X are the same. How should i proceed to ...
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Two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on $X$, and a function $f:X\to Y$ Fréchet differentiable with one of the norms but not with the other one?

There is a theorem that if $f : (X,\|\cdot\|_{X1}) \to (Y,\|\cdot\|_{Y1}) $ is Fréchet differentiable, then replacing the norms with some equivalent norms $\|\cdot\|_{X2}$ and $\|\cdot\|_{Y2}$ ...
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576 views

Dual to the dual norm is the original norm (?)

I have the following questions about dual norms : How do you prove that the dual of the dual norm is in fact the original norm? This is what I have so far: If I have $\|y\|_* $ as the norm dual of ...
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268 views

Low-rank matrix approximation in terms of entry-wise $L_1$ norm

According to the Eckart–Young theorem, the low-rank matrix approximation problem $$\min_{\tilde{A}} \quad \| A - \tilde{A} \|_F \quad \text{s.t.} \quad \text{rank}(\tilde{A}) \le r$$ is given by the ...
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71 views

Example of two norms and ONE linear operator that is bounded and unbounded in a norm.

I am looking for an example of a linear operator that is bounded as well as unbounded depending on which norm you take. Since I do not have much experience with Functional Analysis, I do not know many ...
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Given $\mathbf{A}$ stable (all negative eigenvalues), produce a bound on $\|\mathbf{B}\|$ such that…

Given a system: $\dot{\mathbf{x}}=\left(\mathbf{A}+\mathbf{B}\right)\mathbf{x}$ Can you bound $\|\mathbf{B}\|$ s.t. the origin of $\mathbf{x}$ is exponentially stable using a Lyapunov function? ...
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101 views

Norm space, linear operator exercise, help please!

$f \in L_2[a,b]$ $Uf(s):=\int_a^bk(s,t)f(t)dt$ $k(s,t):[a,b]^2\to R $ continuous. show 1) $U:L_2[a,b]\to L_2[a,b]$, in other words, $Uf(s)\in L_2[a,b] \quad \forall f$ 2) $U$ is linear and ...
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226 views

Poincare Inequality implies Equivalent Norms

I am currently working through the subject of Sobolev Spaces using the book 'Partial Differential Equations' by Lawrence Evans. After the result proving the Poincare Inequality it says the following ...
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51 views

Help in proving that a vector norm satisfies an axiom.

I am trying to prove if the following is a vector norm: ||x|| = max{$|x_1 + x_2|, |x_2 + x_3|, |x_3 + x_1$|} (x is vector with 3 elements) I'm stuck proving that $||\alpha x||=|\alpha|*||x||$. I ...
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1answer
56 views

Calculate norm $1$ of $f(x)=2x^3+3x^5$ belonging to C[-1,1]

Calculate norm $1$ of $f(x)=2x^3+3x^5$ that belongs to $C[-1,1]$. As norm $1$ is called integral norm, I calculated the value of the function for the given interval, and the answer I get is zero. ...
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78 views

Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...
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Asymptotic behavior of $L^2$ norm for increased matrix dimensions

I am playing with matrices which are linear combinations of identity matrix, Pauli spin matrices, $\sigma_x$ and $\sigma_z$ or their tensor products. For example, let the matrix be $H$. So, $H$ could ...