# Tagged Questions

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### $|y-x|\lt \epsilon$ and $|z-x|\lt \epsilon$

I'm trying to solve this question: Is it not obvious we can arbitrarily approximate the points $y$ and $z$ to $x$? but how to formalize this? Any help? hints to begin with? Thanks Notation ...
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### $|b-a|=|b-c|+|c-a| \implies c\in [a,b]$

We know that if $c\in [a,b]$ we have $|b-a|=|b-c|+|c-a|$. I'm trying to prove that if the norm is induced by an inner product, then the converse holds. I need a hint or something. Thanks in advance
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### Characterization of definite positive matrices

We can define a positive definite matrix $A\in M(n\times n)$ as the symmetric matrix where $X^tAX\gt 0$ for every column vector $X\ne 0$ in $n$ coordinates. Suppose $A$ is symmetric, I would like to ...
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### Proof of the cardinality of continuous functions from $[0,1]$ to $[0,1]$.

I've been thinking about the cardinality of continuous functions from $[0,1]$ to $[0,1]$. I know that the cardinality is the same as that of $[0,1]$ and the standard proof using the fact that such a ...
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### Parallel vectors in $\mathbb{R}^n$.

Def: We say that $\vec{x},\vec{y}\in\mathbb{R}^n$ are parallel vectors if $|\vec{x}\cdot \vec{y}|=||\vec{x}||\,| |\vec{y}||$. (i.e equality holds in Cauchy–Schwarz inequality) I'm having some ...
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### A matrix $G$ with all eigenvalues with nonzero real part. Then $t\mapsto |\exp(tG)x |$ is unbounded

I am trying to see why this is true. A book I am reading has this claim without any verification and I'm trying to see why it is true. Let $G$ be an $n\times n$ matrix all of whose eigenvalues have ...
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### Set of sequences -roots of unity

Consider $G_n$ as the multiplicative cyclic group given by the $n^{th}$ roots of unity. $$G_n = \left\{ e^{ 2ik\pi/n} \mid 1\leq k \leq n \right\}$$ Now construct a sequence from each $G_n$ by ...
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### Eigenvalues gone wild

I added some significant details to this problem, as it was apparently not clear to everyone what I want to know: This is a question about convergence of eigenvalues which essentially came up in ...
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### Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
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### Conjunctive Normal Form representation/ First Order Logic.

in my research problem, I need to represent three types of three types of relationships between the variables x,y as the following:: " y Cooperates with x" relationship: means if there is two ...
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### Proving that integrator operator of a kernel satisfies a specific peroperty

I am trying to prove that a integrator operator of a kernel satisfy a specific property say $\phi$. By integrator operator for non-negative definite kernel $\mathcal{K}$ I mean $T_{\mathcal{K}}$ such ...
### Span of Dirac's delta distributions dense in Hilbert space of $L^2$ functions?
According to Wiki a set of elements of a Hilbert space(B) is a basis for that space if: Orthogonality: Every two different elements of $B$ are orthogonal: $⟨e_k,e_j⟩=0$ for all $k$, $j$ in $B$ with ...