Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.
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Can the $0$-norm represent determinism?
In Scott Aaronson's Quantum Computing since Democritus, he presents classical probability theory as based on the $1$-norm, and QM as based on the $2$-norm.
Call $\{v_1,\ldots,v_N\}$ a unit vector ...
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Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers
Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal?
My notes say:
$1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
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0answers
15 views
Subgradient of matrix $l1$-norm
Let $X$ be a square matrix, what is the sub-gradient of $f(X) = ||AX-XB ||_{l1}$?
$A$ and $B$ are both constant matrix.
I am very confuse about the chain rule on matrix derivatives.
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+50
Monotone matrix norms
[Ciarlet 2.2-10]
Let $\mathscr{S}_n$ be the set of symmetric matrices and $\mathscr{S}_n^+$ the subset of non-negative definite symmetric matrices. A matrix norm $\|\cdot\|$ to be monotone if
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1answer
46 views
Difference between maximizing matrix's $l1$ norm and Frobenius norm
The original problem is to maximize the Frobenius norm of a matrix product
$max_{X\in \mathbb{D}}||B^TXA ||_F$, $\mathbb{D}$ is the doubly stochastic matrix set
As this is too difficult due to ...
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1answer
67 views
Proving an alternative norm on Sobolev space is equivalent to usual norm
I have this exercice and my problel is only in item 4, and i will desespere.
Let $f \in L^2(\mathbb{R}^n).$
1- Why the equation $\Delta u - u = \dfrac{\partial f}{\partial x_i}$ admits a unique ...
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Example of two norms on same space, non-equivalent, with one dominating the other
I am looking for an example (with proof) of two norms defined on the same vector space, such that the norms on the two spaces are NOT equivalent, but such that one norm dominates the other...
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1answer
92 views
Linear isometry between $c_0$ and $c$
The following question is an exercise and so I'm just looking for advices and not for answers if it's possible.
I have the following sets in $l^\infty$
$$c_0 := \{x_n \in l^\infty: \lim x_n = 0\} ...
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vote
1answer
21 views
product of bounded linear operators
If I have 2 bounded linear operators $T_1,T_2$ such that $T_2:X\rightarrow Y$ and $T_1:Y\rightarrow Z$. I know that by boundedness, $||T_2(x)||\leq||T_2||\,||x||$ and using the norm of $T$ defined as ...
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2answers
44 views
What is the role of supremum in operator norm
An operator norm is defined as $\|A\|_S=\sup\{\|Av\|:v\in \Bbb R^n, \|v\|=1\}$. Where $\|\cdot\|$ is some norm on $\Bbb R^n$ and $A\in M_n(\Bbb F)$, space of square matrices of dimension $n$ over ...
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1answer
19 views
Linear approximation of matrix norm
Given a square matrix $X=[x_1...x_N]$, and can be vectorized by $y=vec(X)=[x_1^T ... x_N^T]^T$
Is there any linear function can approximate $|| X ||$ (any matrix norm is okay) by using $y$?
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Matrix norm proving problem
I have a homework, one of a few, that I should prove that the norm $||A||_S = sup\{||Av||:||v||=1\}$ Where ||.|| is some norm on $R^n$ into which vector space belongs also $v$. I should mention that A ...
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1answer
36 views
$V$ is finite dimensional iff $V'$ with the weak topology is normable
Why is the following statement valid? Note, $V$ is locally convex Hausdorff topological vector space over $\mathbb{C}$ and $V'$ is the space of all continuous linear maps from $V \to \mathbb{C}$.
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Prove this equality in a Hilbert Space $H$ with Riesz's Representation Theorem.
For $x \in H$, prove that $\sup_{\|z\| = 1} \langle x , z \rangle= \| x \|$
Here is a quick one. If someone could improve this it would be great
Proof
By Cauchy Schwarz, $\langle x,z \rangle ...
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2answers
31 views
norm of inverse less than 1
I just wanna ask if what I am doing here make sense:
Let $\epsilon$ be arbitrary positive number. Choosing $\epsilon$ and let it approaches 0, I would like to have $||(I-\epsilon A)^{-1}|| < 1$. ...
