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

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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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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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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 ...
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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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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 ...
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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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25 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 ...
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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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$\kappa(B^{-1}A)=\kappa (AB^{-1}) = \frac{\lambda_{\max}(B^{-1}A)}{\lambda_{\min}(B^{-1}A)}$

Prove or disprove: if $A,B$ are symmetric positive definite (s.p.d.) matrices then operator 2-norm condition number $\kappa(B^{-1}A)=\kappa (AB^{-1}) = ...
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$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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Is the matrix norm of a matrix equal to the maximum of the norms of its Jordan block?

Let $J$ be a Jordan block matrix with blocks $J_1,\cdots,J_n$. I came up with some examples of $J$ and noticed that $\|J\|=\max_{i=1,\cdots,n}\|J_i\|$. Does this result always hold? The norm I use ...
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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 ...
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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 ...
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21 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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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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25 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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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 ...
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Question on projections in linear algebra

Hello all I was given this question for linear algebra class which I am stuck on and would truly appreciate the help: $V$ is a finite-dimensional inner product space with $M$ and $N$ non trivial ...
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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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Change in singular values of matrix after left-multiply with a diagonal matrix

Say that we have an SVD for a matrix $X = U \Sigma V^T$, giving trace norm $||X||_{tr} = ||\Sigma||_{tr} = \sum \Sigma_{ii}$. I am wondering what happens to the SVD and/or trace norm if we left ...
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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} ...
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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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Derivative of squared norm of a complex function

Assuming $f(x)$ is a complex function, its squared norm is defined as $$|f(x)|^2 = f(x) . f^*(x)$$ What is derivate of $|f(x)| ^ 2$?
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Calculate the derivative of a complex norm

I'm stuck with a rather trivial looking question. How do you calculate the derivative of the norm of a complex number to it self? Like in $$ \frac{d|a|^2}{da} = ? $$ I think it would give rise to a ...
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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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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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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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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: ...
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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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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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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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[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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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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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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The norm of a 4-vector

Maybe this is more of a physics question, but it's about manipulating 4-vectors, so I thought maybe it could go here. Let me know if I should post in physics instead. The norm of the momentum 4-vector ...
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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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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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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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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,\ \ ...