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

learn more… | top users | synonyms

0
votes
1answer
16 views

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 ...
1
vote
2answers
42 views

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 ...
0
votes
1answer
24 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 ...
1
vote
0answers
12 views

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) = ...
1
vote
0answers
45 views

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 ...
0
votes
0answers
9 views

Normalize and Average weighted

Everyday I receive a data of three variables (neutral, negative and positive). ...
0
votes
0answers
25 views

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 ...
2
votes
0answers
20 views

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 ...
0
votes
1answer
20 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, ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
2
votes
1answer
43 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 ...
3
votes
1answer
19 views

“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 ...
2
votes
1answer
22 views

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 ...
0
votes
0answers
22 views

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} ...
0
votes
0answers
36 views

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 ...
8
votes
2answers
170 views

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 ...
3
votes
1answer
31 views

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 ...
0
votes
1answer
27 views

$L^p$ 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? $$\|fg\|_p\leq\|f\|_p\|g\|_p$$
0
votes
0answers
6 views

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 ...
0
votes
0answers
25 views

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} ...
0
votes
1answer
19 views

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) ...
1
vote
1answer
25 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: ...
1
vote
3answers
42 views

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?
1
vote
1answer
16 views

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 ...
0
votes
1answer
28 views

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 ...
1
vote
1answer
44 views

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 ...
5
votes
1answer
40 views

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 ...
0
votes
0answers
15 views

[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 ...
2
votes
1answer
26 views

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 ...
0
votes
0answers
15 views

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 ...
0
votes
0answers
13 views

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 ...
0
votes
1answer
38 views

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 ...
0
votes
1answer
14 views

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 $A$ ...
0
votes
0answers
13 views

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 ...
3
votes
0answers
39 views

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 ...
0
votes
0answers
54 views

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). ...
1
vote
0answers
25 views

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,\ \ ...
4
votes
1answer
47 views

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 ...
5
votes
2answers
98 views

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 $$||z||_*= \sup \{ z^Tx : ||x|| \leq 1 \}$$ ...
0
votes
0answers
9 views

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 ...
3
votes
2answers
29 views

Continuity of matrix product with respect to matrix norm?

I'm trying to teach myself about ordinary differential equations with an old script and I'm struggling with this problem: Show that the matrix product is continuous with respect to the matrix norm. ...
-1
votes
1answer
12 views

What does it mean to define a norm as $\|.\|_{A, B}$? (e.g. $\|.\|_{\infty, [-1, 1]}$)

I have seen when many instances of $\|.\|_{A}$, e.g. $A = \infty$ or $1$, etc. What does it when we write a norm as $\|.\|_{A,B}$? For example see theorem 3 of this, which uses $\|.\|_{\infty, [-1, ...
-1
votes
0answers
29 views

norm of functional defined on subspace of $C[0,1]$

Denote $X=\lbrace f\in C[0,1];3\int_0^1f(x)\;dx=4\int_0^1xf(x)\;dx,\;\parallel f \parallel=\max_{[0,1]}|f(x)|\rbrace$ and $F:X\rightarrow \mathbb{R}$, $F(f)=\int_0^1f(x)\;dx$. What is the norm of $F$ ...
1
vote
0answers
16 views

Find $\langle f,g \rangle$ w.r.t. $L_0 \perp L_1$.

Let $X=C[-1,1]$, and $L_k= \{ <t^{k+2i}, i=0,1,2,... > \} $. Define an inner product on $X$ with respect to $L_0 \perp L_1$. Then confirm that $L_0 \perp L_1 $ on your inner product. Can we ...
0
votes
1answer
45 views

Is $X$ one dimensional

If $X $ is an inner product space and if there exists $x \in X $ s.t. $\{x\}^{\perp}=0$. Is $X$ one dimensional? The way I have written out this question a bit wrong because it was in my exam so I ...
0
votes
2answers
26 views

Is $T(\ell^1 ) \subseteq \ell^1$?

If we have a linear operator $(\ell^{\infty} , \|\cdot \|_{\infty} ) \rightarrow (\ell^{\infty} , \|\cdot \|_{\infty} ) $ by $T((a_k)_{k \ge 1}) = (b_k)_{k \ge 1}$ where $$ b_k= ...
0
votes
0answers
38 views

Derivative of L2 norm

I am reading a paper about image processing and I have a question. In the paper we have equations like below. $X_{C1} = 0.596X_R - 0.274X_G - 0.322X_B$. $X_{C2} = 0.211X_R - 0.523X_G + 0.312X_B$ ...
2
votes
1answer
40 views

monotonic decrising p norm

I have the following inequality I need to prove: $$\|x\|_q \leq \|x\|_p \leq n^{\frac{1}{p}-\frac{1}{q}}\|x\|_q$$ For the right inequality, how can i prove $\|x\|_p$ is monotonically decreasing ...
0
votes
1answer
45 views

Prove or disprove: If $1=||A||>||B||$, then $A-B$ is nonsingular.

Prove or disprove: If $1=\|A\|>\|B\|$, then $A-B$ is nonsingular. I think that since $\|A-B\|>0$ by the given conditions we know it is nonsingular. Any solutions or hints are greatly ...