All Questions
Tagged with upper-lower-bounds normed-spaces
60
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The lower bound of Frobenius norm of matrices product.
Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, where $\|.\|$ is Frobenius norm.
I would like to find the lower bound of $\|A^{-1}.B.(A+B)^{-1}\|$:
I have the result:
$$\|A^{...
- 11
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52
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Greatest lower bound and smallest upper bound of Frobenius norm of product of matrices
Let $A,B \in M_p(R)$ be symmetric matrices, $A$ is given and non-singular, $\|A^{-1}\|.\|B\| < 1$ where $\|.\|$ is Frobenius norm.
I would like to find the lower bound and upper bound of $\|A^{-1}....
- 11
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75
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Lower bounding $v^TA^{\dagger}v$ where A is a sum of rank-1 matrices
This is related to a previous question I asked (Upper bounding $v^TAv$ where $A$ is the inverse of a sum of rank-$1$ matrices and $v$ is a vector).
Let $(x_i)_{1 \leq i \leq n}$ be vectors of $\...
- 182
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1
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172
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Lower bound of norm squared
Let $(X, \| \cdot \|)$ be an $n$-dimensional normed
$\mathbb{R}$-linear space. Let $\{x_1, x_2, ..., x_n\}$ be a basis of $X$. Show
that there exist positive constants $C_1$ and $C_2$ such that for ...
- 580
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342
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Bound on L2-Norm of Probability Distributions?
I am working on a problem where I have continuous probability distributions $p$ over a bounded domain $D$, i.e., $\forall x$ $p(x) \geq 0$ and $\|p\|_1 = \int_D p(x) dx = 1$. However, I also want $p$ ...
- 127
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How to find an upper bound for a norm with power less than 1?
I need to find an upper bound for $$\|x-y\|_2^\nu$$ in which $\nu \in [0,1]$. I was wondering if $\|x-y\|_2^2$ can be considered as an upper bound. or $\|x\|+\|y\|$?
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If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?
If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?
If is not true in general, please give some counter-...
- 1,356
11
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323
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Can a norm on polynomials be supermultiplicative?
A norm on a real algebra is supermultiplicative when $\lVert f\cdot g\rVert\geq\lVert f\rVert\cdot\lVert g\rVert$ for all $f$ and $g$ in the algebra.
Is there a supermultiplicative norm on $\mathbb R[...
- 5,681
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255
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Can a norm on polynomials be "almost multiplicative", even for large degrees?
Definition: A norm on a real algebra is called almost multiplicative if there are positive constants $L$ and $U$ such that, for all $f$ and $g$ in the algebra,
$$L\lVert f\rVert\cdot\lVert g\rVert\;\...
- 5,681
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481
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Upper bound on the norm of an inverse of a positive definite matrix
Given two real-valued symmetric positive-definite matrices $A$ and $B$. Assume that $A\succeq I$. Let $\Delta = B-A$. We are interested in a bound on $\Vert B^{-1}-A^{-1}\Vert$ in terms of $\Vert \...
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421
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Upper Bound for the Induced 2-Norm $\|(A+B)^{-1}A\|$
Assume $A$ is a positive definite matrix, and $B$ is a positive semi-definite matrix. I am interested in the problem of whether there exists a constant upper bound for the induced 2-norm (spectral ...
- 163
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199
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supremum on quadratic and bilinear form
Let $V$ be a real normed vector space. Denote by $V_1$ the subset of $V$ of vectors with norm $\le 1$. Consider a positive-semidefinite symmetric bilinear form $\langle \cdot, \cdot\rangle : V\times V\...
- 8,107
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334
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bounds on norm of a vector
Let $x \in \mathbb{R}^n$, and $M \in \mathbb{R}^{n\times n}$ be a full rank square matrix. If it is known that $ \| M x\|_2 \leq c$, then what can be said about the upper bound of $\| x\|_2$, i.e., $\|...
- 1,722
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72
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Lower bound on the components of a unit vector
Let $u \in \mathbb{R}^n$ be a unit vector: $\|u\| = 1$. Is there a formula for the smallest $C > 0$ such that $|u_i| \geq C$ for some component $u_i$ where $i = 1,...,n$?
For example, if $u \in \...
- 1,093
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If the norm of the difference between two unit vector is lower bounded by a positive constant, does it mean that the inner product is upper bounded?
Let $x,y$ be two vectors with $\lVert x \rVert = \lVert y \rVert =1$ and $\lVert x-y \rVert \geq \delta$, where $\delta \gt 0$. Is it possible to show that, $1-(x^Ty)^2 \geq \delta^2$?
My Approach:
$$\...
