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

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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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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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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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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 ...
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52 views

Unit ball of $L^1$, $L^\infty$ and $C(X)$ is not strictly convex

I need to show that the unit balls of $L^1(\mu)$, $L^\infty(\mu)$ and $C(X)$ are not strictly convex. I have already shown that if $1<p<\infty$ then the unit ball of $L^p(\mu)$ is strictly ...
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Properties of unit vector scaling

What properties are kept when we scale a vector to unit length, i.e. $\frac{\mathbf{v}}{||\mathbf{v}||_1}$? Imagine that we have an unconstrained optimization problem, and we obtain as solution $x_i ...
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Prove that $K(u) = ||u||Mu+f$ is a contraction

Let $M$ be a $2 \times 2$ real matrix such that $$\parallel{}Mx\parallel{} \leq \frac{1}{4}\parallel{}x\parallel{}$$ for all $x \in \mathbb{R}^2$ where $\parallel{}\parallel{}$ is the euclidean norm ...
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Norms on $\mathcal{P}_N$ Vector Space of Polynomials up to Order N

$\|p\|_\infty :=\sup_{x\in [0,1]}|p(x)|$ and $\|p\|_{L^1}:=\int_0^1 |p(x)| dx$. As the space of real-valued polynomials on $[0,1]$ up to order $N$ is a $N+1$ dimensional vector space and $\|\cdot\...
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Triangle inequality Trace norm

when becomes the triangle inequality for the trace norm an equality? I search for it in books and web, but couldn´t find it. Thanks for help!
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Convex set in a vector space gives a norm

Given an $\mathbb{R}$ or $\mathbb{C}$ vector space $X$ and a function $p:X\rightarrow[0,\infty)$ with $p(x)=0$ iff $x=0$ and $p(\alpha x)=|\alpha|p(x)$ for all $x,\alpha$, I want to show that $p$ is a ...
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Prove or disprove continuity of two maps

Yet another time I need help to prove continuity of a certain map and don't know how to do it: Look at the vector space $$C_b^1(\mathbb R; \mathbb C) := \{f \in C^1(\mathbb R;\mathbb C):||f||_{\...
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30 views

Vector norm lemma and proof

I have a question from Numerical linear algebra book by Trefethen & Bau : Let $\|\cdot\|$ denote any norm on $C^m$. The corresponding dual norm $\|\cdot\|'$ is defined by the formula $\|x\...
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Inequality with derivative and supremum norm

I have the following property written in a book but I can't understand why this implication is true. I would be glad if anyone could help me let $A \in \mathbb{R}^N$. $$\frac{d}{dt} \nabla A(x,t) = \...
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Dual Norm proof

Let $\|.\|$ denote any norm on $C^m$. The corresponding dual norm $\|.\|'$ is defined by the formula $\|x\|' = sup_{\|y\|=1}|y^*x|$. (a)Prove that $\|.\|'$ is a norm? (b) Let $x, y \in C^m $ with ...
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Normalize and Average weighted

Everyday I receive a data of three variables (neutral, negative and positive). ...
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Derivative of Frobenius norm expressions

For an optimization problem using the L-BFGS algorithm, I am trying to use the gradients of two norm expressions. X are matrices, x elements of X. $$R_a = \Lambda * \sum_{c=1}^C ||X_c - 1/C \sum_{c=1}...
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Norm of infinite dimensional Hilbert space to calculate difference between string lengths

I am trying to wrap my head around Proposition 13, last para, page 1049 in this paper. The authors are trying to prove certain properties of string edit distance (defined at the start of Section of 6....
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1answer
24 views

“Length” function without absolute homogeneity property?

A norm $\|\cdot\|$ must have the property of absolute homogeneity. I'm working with a function that acts like a "length," but which can also include negative numbers (so "length" is used loosely here, ...
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19 views

Proving vector norm

Quite unsure about this problem. Prove that for vectors $u, v ∈ R^n$ we have $$\Vert u + v\Vert^2 +\Vert u − v\Vert^2 = 2 \Vert u\Vert^2 + 2 \Vert v \Vert ^2$$ Can you just expand the left hand part ...
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26 views

Finding the closest element to a function in a normed space containing functions.

Let $B=\{f\in c[1,0]|\forall 0\leq x\leq 1 : f(x) \geq 0\}$. Given $f\in c[0,1]$, find the closest element in $B$ under the $\|\cdot\|_2$ norm. I can see something similar to this question in the ...
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49 views

Is the converse of the Pythagorean Theorem false for complex inner products?

I was thinking about the converse of the Pythagorean theorem: $\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y\rVert^2 \implies x \perp y$ Does this hold if the inner product $\langle \cdot,\...
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“Reverse” of frobenius matrix norm inequality

Suppose that we have some $m \times n$ matrix $C$, and its full rank (skeleton) decomposition $$ C = AB^T, $$ where $A$ is $m\times r$ and $B$ is $n\times r$ for some $r$. We know that frobenius norm ...
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Contraction Banach theorem

Given the following function: $$g(z)=C*\begin{pmatrix} x^2+y^2-2 \\ x^2-y^2-1 \\ \end{pmatrix}+z, \; \; \;z=(x,y)\in [0.93,1.52]\times [0.41,1]$$ Prove that $g $ ...
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Find the norm of a linear functional in $L^2[0,1]$.

Define the linear functional $f : L^2[0,1] \text{(As completion of $C[0,1]$, all the continuous complex-valued function )} \mapsto \mathbb{C}$ by $$f(\psi)=3\int_{0}^{1}\psi(t)dt + i\int_{0}^{1} \psi(...
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Proof that $\sum_{j=0}^\infty C_j$ converges if $\sum_{j=0}^\infty \|C_j\|$ converges

$C_j$ is a sequence of matrices in $\mathbb C^{n \times n}$ and the identity $$\max_{j,k}|A_{j,k}|\leq \|A\|\leq n\max_{j,k}|A_{j,k}|$$ is known. Show that $\sum_{j=0}^\infty C_j$ converges if $\...
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Prove of inequality under a Hilbert space.

Let $x\neq y$ when $x,y\in H$ and H is a Hilbert space which satisfy $\|x\|=\|y\|=r$. Show that $\|\frac{x+y}{2}\|<r$. Actually in my question r=1 but as far as i could understand there is a way ...
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Norm equivalence on $l^1$.

Suppose that $\|\cdot\|$ is a norm on $l^1$ such that: a) $(l^1, \|\cdot\|)$ is a Banach space, b) for all $x \in l^1$ $\|x\|_{\infty} \leq \|x\|$. Prove that the norms $\|.\|$ and $\|.\|_1$ are ...
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$L^p$ Norm of product of two bounded functions

If $f$ and $g$ are bounded functions in $L^p[a,b]$, does the following inequality hold in $L_p$ spaces? $$\|fg\|_p\leq\|f\|_p\|g\|_p$$
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Spinor norm of the pth power of a matrix

Let $F_{q}$ be a finite field of order $q=p^{r}$ ($p$ odd) and let $V$ be a $3$-dimensional vector space over $F_{q}$. Consider the subgroup $\Omega(3,q)$ of $SO(3,q)$., where we are picking the ...
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Is complex multiplication the only multiplication operation on $\mathbb{R}^2$ that works with the Euclidean norm?

What I'm asking is: viewing complex multiplication as binary operation on $\mathbb{R}^2$, is usual multiplication of complex numbers the only operation $\otimes$ on two vectors $\vec{u}$ and $\vec{v} \...
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Length of a curve under a non-Euclidean norm in the integral form.

Let $V$ be a normed space. Let $\gamma\colon [a,b] \rightarrow V$ be continuous. Then $\gamma$ is a curve. Let $P$ be a partition of $[a,b]$, then $$\Lambda(\gamma, P) := \sum_{i=1}^n \| \gamma(x_i) -...
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27 views

norm from inner product

I have a question in a Hilbert Spaces course as follows: Let $X=(x_1, x_2)$ be vector in a vector space of all ordered pairs of complex numbers X. Can we obtain the norm defined on X by: $\|X\|=|x_1|...
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Find a matrix $A \in \mathbb{R}^{2 \times 2}$ such that $ \|Ax\|_{2}=\|x\|_{2}$ for every $ x\in \mathbb{R}^2 $

How to find a matrix $A \in \mathbb{R}^{2\times 2}$, $A\neq I_{2}$ such that for every $ x\in \mathbb{R}^2$ we have $\|Ax\|_{2}=\|x\|_{2}$. Is that even possible?
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Which one is the correct definition of natural norm?

In the definition 2 of Normal Subgroup Reconstruction and Quantum Computation Using Group Representations, the authors have defined the natural norm of a matrix as follows. The natural norm of the ...
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Why the separate notation for norm

One usually denotes the norm as $\|\cdot\| $, $\| v\| := \sqrt{\langle v, v \rangle}.$ However, in metric spaces, one often writes $d(x,y) \equiv \lvert x-y \rvert$. Since the norm canonically ...
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Applaying equivalence of norms to show a sequence is a Cauchy sequence

Let $\|\cdot\|$ be any norm on $\mathbb R^n$ prove that a sequence $x \in \mathbb R^n$ is a Cauchy sequence under $\|\cdot\|_2$ if and only if it is a Cauchy sequence under any $\|\cdot\|$. I tried ...
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Norms inequality in a sequence space

Let $1 \leq p<q \leq \infty$ (p an q are not related) Let $\Phi$ be the vector space of all sequences with at most finitely many nonzero elements, meaning $\Phi=\{\{x_n\}_{n=1}^\infty|$ there is $...
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[svm]A Problem of max(1/|w|) equal to min(1/2*|w|^2)

I've been search many SVM theory thesis for machine learning Those articles usually say max(1/|w|) equal to min(1/2*|w|^2) but they didn't write the detail of the mathematics process. I also read this ...
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Norms Equivalence over $\mathbb R^n$

Let $\|\cdot\|$ be any norm on $\mathbb R^n$. Prove that a sequence in $\mathbb R^n$ is Cauchy under the $\|\cdot\|_2$ norm if and only if the sequance is Cauchy under the $\|\cdot\|$. Prove that a ...
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How to measure alignment of a set of complex numbers

Consider a vector of complex numbers $v=(z_1,z_2,\dots,z_n)$ with $||v||=v^*v=1$. Each of the components $z_i$ represents a point in the complex plane, and all these points can be represented in the ...
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How can we prove that Unitary Transformation is isometric?

I am studying Image Processing in which unitary transforms play an important role, one reason that I found for their use in image transformation is isometry(they preserve distance), I found a relevant ...
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Almost negative definite matrices and norm-distance matrices

An "almost negative definite" matrix $A$ satisfies the property $$ v^te = 0\implies v^tAv\le 0 $$ where $e=(1,1,\dots,1)$. We know that if $A$ is a simmetric zero-diagonal (hollow) matrix, then $A$ ...
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Is the norm a proper convex function?

Norms are closed convex functions. Are they also proper convex functions? Let $\Vert \cdot \Vert$ be a norm. To prove it is proper, it is sufficient to say that, since $$ \Vert 0 \Vert = 0$$, then ...
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Matrix with roots of unity entries

For a given prime p, i am interested in the norms of matrices which have root of unity entries, i.e., $M_{k,l} \in \{1, \zeta, \dots, \zeta^{p-1}\}$ where $\zeta = \exp{(2\pi I/p)}$. Are there any ...
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Can the equation $x+y+z=1$ describe a sphere?

I know that in a three-dimensional Euclidean space, with the Euclidean distance, $x+y+z=1$ describes a plane. In the same conditions, $x^2+y^2+z^2=1$ would be a sphere (a 2-sphere to be exact). ...
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Find norm of $T:(\ell^1,||\cdot||_1)\to(\mathcal C[0,1],||\cdot||_\infty),$ $(T(\xi))(x)=\sum_{k=0}^\infty a_k\xi_k x^k,$ $\xi\in\ell^1$

Let $a=(a_0,a_1,\cdots)$ be a fixed element of $\ell^\infty$. Define $$T:(\ell^1,\lVert\cdot\rVert_1)\to(\mathcal C[0,1],\lVert\cdot\rVert_\infty),\ (T(\xi))(x)=\sum_{k=0}^\infty a_k\xi_k x^k,\ \ \...
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Applying equivalence of norms on $\mathbb R^n$ .

Let $\|\cdot\|$ be any norm on $\mathbb R^n$. Prove that a sequance on $\mathbb R^n$ converges to an element $x \in \mathbb R^n$ under the $\|\cdot\|_2$ norm if and only if the sequance converges to $...
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Equivalent characterizations of the dual norm on finite dimensional vector spaces

In their book on Convex Optimization, Boyd and Vandenberghe state that given a norm, $||\cdot||$, defined on $\mathbb{R}^n$, the dual norm is defined as $$||z||_*= \sup \{ z^Tx : ||x|| \leq 1 \}$$ ...
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Minimizing matrix norm via left-multiplication by $SL(m)$

Suppose that $M$ is an $m\times n$ matrix of full row rank, with $m \leq n$. Then if $\|M\|$ is the matrix norm induced on $M$ from the norm on our vector space, we can look for the following ...