All Questions
Tagged with upper-lower-bounds normed-spaces
60
questions
2
votes
1
answer
98
views
Show that $\sum_{k=0}^{N} |P(k)| \leq C(N) \int_{0} ^{1} |P(t)| dt $. [closed]
I’m attending a functional analysis course and I am given to solve this problem as an exercise but I’m a little bit disoriented and I don’t know what tools I can use to get it.
Show that, for each $...
- 251
0
votes
0
answers
651
views
Lower bound for $\|x-y\|$
For $x$,$y$ in Hilbert space $\mathcal{H}$ I want a lower bound for
\begin{equation}
\|x-y\|_{\mathcal{H}}^2
\end{equation}
I know
\begin{equation}
|\ \|x\|_{\mathcal{H}}-\|y\|_{\mathcal{H}}\ |\leq\|...
- 142
0
votes
1
answer
76
views
Coercive bilinear form for maximum norm
Let $f$ be a differentiable function. Denote a bilinear form by $$b(f,f) = \int_{0}^{1} \bigg( \frac{d f(x)}{dx} \bigg)^{2} dx.$$ Given $f(0) = 0,$ we want to show that $$a \cdot b(f,f) \geq ||f||_{\...
- 802
0
votes
1
answer
96
views
How to bound this function?
I am trying to show that the derivative of the scalar function $V(\mathbf{x})$, $V'(\mathbf{x})$, is such that
$V'(\mathbf{x})=-(x_1^2+x_2^2)+x_2u\leq-||\mathbf{x}||^2+||\mathbf{x}|| \ |u|$
where $\...
- 3
1
vote
1
answer
859
views
A lower bound for the condition number matrix
I have the following proposition:
Theorem: For every invertible matrix $A\in\mathbb{R}^{n\times n}$ and every matrix norm $\|\cdot\|$, then the condition number $\mathcal{K}(A):=\|A\|\cdot\|A^{-1}\|$ ...
- 349
0
votes
1
answer
43
views
Mean of a function, under different distributions
Define a function: $f: X \rightarrow \mathbb{R}$. The notation $\mathbb{E}_{x \sim D}[f(x)]$ denotes the mean of the function under distribution $D$, where $D$ is a continuous distribution defined ...
- 2,670
1
vote
1
answer
46
views
Condition for $\|X_n^{-1}\|$ to be bounded?
Let $\lambda_n = \|X_n^{-1}\|$, where $X_n$ is a non-singular $p\times p$ square matrix and $\|A\| = \sup_{|x| = 1}|Ax|$, with $|\cdot|$ the Euclidian norm. Is there a sufficient condition so that the ...
- 882
0
votes
1
answer
32
views
Is there an $L<1$ such that $\left\Vert f(z_1) -f(z_2) \right\Vert_2 \leq L \left\Vert z_1 -z_2 \right\Vert_2 $ [closed]
Let $f(x,y)=(\sqrt{4-y^2},\sqrt{1-\frac{x^2}{16}})$. I'm trying to show if there exists an $L<1$ such that
$\left\Vert f(z_1) -f(z_2) \right\Vert_2 \leq L \left\Vert z_1 -z_2 \right\Vert_2 $ for ...
- 1,669
1
vote
1
answer
2k
views
Upper bound on $L_\infty$ norm of product of matrices
Let $M_1$ and $M_2$ be two $n \times n$ matrices. Suppose, $||M_1||_\infty \leq U_1$ and $||M_2||_\infty \leq U_2$. What is the upper bound on $||M_1.M_2||_\infty$? Here is my analysis.
$||M_1.M_2||_\...
- 145
0
votes
0
answers
123
views
Does an each element of sum of bounded function is bounded?
I have a sum of two element as following:
$$e_i=x_{i-1}-x_i$$
I know that $e_i \in L_{\infty}$ so it is bounded. Is it true then that $x_i$ and $x_{i-1}$ are also in $L_{\infty}$ (bounded)?
- 25
0
votes
1
answer
103
views
Sharp Lower Bound for Entrywise 1-norm of a Real Semi-Orthogonal Matrix
Let $A$ be a real-valued $m\times n$ ($m>n>1$) matrix such that $A^TA=I$, what is the sharp lower bound for $\|A\|_1=\|\operatorname{vec}(A)\|_1=\sum|A_{i,j}|$?
Since one can show the ...
- 823
4
votes
0
answers
379
views
Need to improve upper bound for $\| (uv^T + B)^{-1} \|$ (Sherman-Morrison formula)
I have a matrix $A \in \mathbb{C}^{n \times n}$ in the form $A = uv^T + B$, where $u,v \in \mathbb{C}^n$ and $B \in \mathbb{C}^{n \times n}$. I know $A$ is invertible and want to find an upper bound ...
- 6,574
1
vote
1
answer
110
views
Norm of elements in discrete Heisenberg group
The discrete Heisenberg group (https://en.wikipedia.org/wiki/Heisenberg_group#Discrete_Heisenberg_group) is generated by
$
x=\begin{pmatrix}
1 & 1 & 0\\
0 & 1 & 0\\
0 & 0 & ...
user276611
2
votes
2
answers
673
views
If the expectation of the norm of random variable is bounded, does that imply that the expectation of the squared norm is bounded?
Let $\mathbf{x}$ be a random vector. If $\mathbb{E}[\|\mathbf{x}\|]$ is upper bounded, does that imply that $\mathbb{E}[\|\mathbf{x}\|^2]$ is also upper bounded?
The Jensen's inequality goes the ...
6
votes
1
answer
6k
views
Matrix Norm Bounds
A natural consideration of matrix norms is to compare them, and one of the many standard results on the induced 1, 2, and $\infty$-norms indicate that
$$\frac{1}{\sqrt{n}}\|A\|_\infty\leq \|A\|_2\leq \...
- 1,793