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1 vote
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Boundedness of a matrix operator in a norm

As explained in the comments, you are mixing up two different concepts: boundedness of a matrix and uniform boundedness of a family of matrices. To illustrate this via the example \begin{align*} A:(0,\...
Frederik vom Ende's user avatar
1 vote

Bounding $\Vert f\Vert \Vert g\Vert$ by $\Vert wf \Vert^2 +\Vert w^{-1} g\Vert$

The inequality does not hold for any nonconstant $w(x).$ Assume $f$ does not vanish a.e.. For $d=1$ the LHS is equal $$\int\limits_0^1[w^2(x)|f(x)|^2+w^{-2}(x)|g(x)|^2]\,dx\ge 2\int\limits_0^1 |f(x)|\,...
Ryszard Szwarc's user avatar
2 votes
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Bounding $\Vert f\Vert \Vert g\Vert$ by $\Vert wf \Vert^2 +\Vert w^{-1} g\Vert$

For $f = \frac{1}{w}, g = w$ the left hand side is just $2$, so we only have to pick $w$ such that the right hand side is larger than that. For example when $d = 1$ for any $n \in \mathbb{N}$ we can ...
user23571113's user avatar
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1 vote
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Convex combination of equidistant curves

This is untrue. Consider $\gamma_0(t) = (t,0)$, $\gamma_1(t) = (t-1,0)$ and $\gamma_2(t) = (t- \cos t, -\sin t)$, for $t \in \Bbb R$. Then \begin{align} \gamma_0(t) - \gamma_1(t) &= (1,0), \\ \...
Didier's user avatar
  • 20k
0 votes

Strong smoothness of Lp norm

In Example 5.11 of Beck(2017), he has proved the (p-1)-smoothness of the half-squared $l_p$-norm function for $p\in[2,+\infty)$. I think that following the derivation, we can get the $1$-smoothness ...
Min Xu's user avatar
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1 vote

Why do variances add when summing independent random variables?

The other answers explain how the additivity of variance for sums of independent random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ follows from orthogonality/Pythagoras' ...
JKL's user avatar
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2 votes

Why do variances add when summing independent random variables?

I think you can indeed think about it in terms of Pythagorean theorem. So the point is that you can think of random variables as vectors in some space, so the variance of a random variable is like the ...
Alan Chung's user avatar
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2 votes
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Which metrics (on vector spaces) can be induced?

If $d$ is induced by a norm then $\|x\|=d(0,x)$ is that norm. So we immediately have the conditions: $d(0,\lambda x)=\lambda d(0,x)$ $d(x,y)=d(0,y-x)$ From this, it follows that $\|\cdot\|$ ...
Zoe Allen's user avatar
  • 5,603
2 votes
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Isomorphism between linear functionals $\mathbb{L}(\mathbb{R}^2)$ and $\mathbb{R}^2$.

Using the suggestion of Lagrange Multiplier Method of @AlvinL. We can represent it by a $2 \times 1$ matrix representing its coefficients. Let F(x, y) = ax + by , then the p-norm is \begin{align*} ...
Wellington Silva's user avatar
0 votes

If a pseudonorm function $N$ is continuous in a given topology, does the pseudometric topology formed by $d(x,y)=N(x-y)$ coincide with first topology?

No. In Furstenberg's topology $\phi$ on $\Bbb{Z}$ in particular. The open balls formed by the norm would look like: $$ B_r(y) = \{ x \in \Bbb{Z} : N(y-x) \lt r\} = N(y-\cdot)^{-1} \{1, \dots, \...
SeekingAMathGeekGirlfriend's user avatar
1 vote
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Triangle inequality for $l^2$ norm

Let $T:Z\oplus U\to \mathbb{R}^2$ be defined by $T(z,u)=(\|z\|,\|u\|),$ where $\mathbb{R}^2$ is equipped with $\ell^2$ norm. Then $$\|T(z+z',u+u')\|_2\le \|T(z,u) +T(z',u')\|_2\\ \le \|T(z,u)\|_2+\|T(...
Ryszard Szwarc's user avatar
0 votes

Derivation of subgradient of a matrix's nuclear norm

When you have a nice set of orthogonality properties like $U^\mathsf{T}W = 0, WV = 0$ and you are asked to analyze a sum of form $UV^\mathsf{T}+W$, you should try expressing everything as a block ...
Alex Nguyen-Le's user avatar
0 votes

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

The rank constraint is not convex. Still one could borrow some ideas from Convex Optimization. If one define the problem as: $$\begin{align} \arg \min_{ \boldsymbol{X} } \quad & {\left\| \...
Royi's user avatar
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0 votes

Norm preservation properties of a unitary matrix

Let $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ be either the field of real numbers $\mathbb{R}$ or the field of complex numbers $\mathbb{C}$. Definition (Unitary matrix). A unitary matrix is a square ...
Steven van Dokkum's user avatar
3 votes

Metric on $\mathbb R^{m \times n}$ based on the Frobenius norm

$\def\R{\mathbb{R}}\def\T{^{\mathrm{T}}}\def\ge{\geqslant}\def\le{\leqslant}\def\peq{\mathrel{\phantom{=}}}\def\norm#1{\left\lVert#1\right\rVert}\def\paren#1{\left(#1\right)}$Proposition 1 (Ptolemy): ...
Ѕᴀᴀᴅ's user avatar
  • 34.8k
2 votes
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For a given $k \in \mathbb{N}$, define the mapping $A: X \to X$ by the rule $ (Af)(x) = x^k f(x). $

To show that $A$ is linear means to show that $$ A(\alpha f+\beta g)=\alpha Af+\beta Ag $$ or, equivalently, that $$ x^k(\alpha f(x)+\beta g(x))=\alpha x^kf(x)+\beta x^kg(x). $$ This identity clearly ...
John B's user avatar
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0 votes
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I want to prove that $T$ belongs to $B(X)$, meaning it is a bounded linear operator.

I think that $f \in X^*$ should be a bounded linear operator as well. I'm assuming it is a operator of this kind: $f:X\to\mathbb{K}$ where $\mathbb{K}$ is a field equal to $\mathbb{R}$ or $\mathbb{C}$...
Alejandro Sánchez Yalí's user avatar
2 votes

Metric on $\mathbb R^{m \times n}$ based on the Frobenius norm

The following is a partial answer as it only fully addresses your second question. As for the first question: while numerics suggest that your map $d$ is indeed a metric, I cannot even prove the ...
Frederik vom Ende's user avatar
6 votes
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Nature of the Euclidean Norm

This is subtle. In mathematics we have the freedom to define distances however we want depending on the application, and for example sometimes rather than the Euclidean norm we use what are called the ...
Qiaochu Yuan's user avatar
1 vote

Nature of the Euclidean Norm

The definition is partly "an greed-upon convention", but at the same tame there is a reason to define it the way it is. There is mathematical notion called "metric" or "...
César VB's user avatar
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