# Tagged Questions

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### prove that F is dense in C(X×Y,R)?

Let X,Y be compact metric spaces. Let F= {∑ Ai fi(x) gi(y),fi∈∁(X,R),gi∈∁(Y,R), i from 1 to n }. prove that F is dense in C(X×Y,R) ? please i cant figure it out any help i will be thankful !
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### Non-Orthogonal Eigenvectors and Computation?

Say for a real, rectangular matrix $X$ and a s.p.s.d matrix $Q$ we maximize or minimize $Tr(X^TQX)$ under the constraint $Tr(X^TM) = 1$ for some fixed real matrix $M$. i) Would the columns of the ...
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### set of symmetric positive definite matrix open?

I consider a collection of symmetric positive definite matrices of the same dimension. I've learned it's an open set but have no clue about the proof. Also, can the symmetry condition be dropped? ...
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### Operator norm of real matrix

I've been looking through my workbook in preparation for the next set of classes and I'm stuck on this problem and don't know how to possibly proceed with it. The hint isn't helping and there isn't ...
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### Do non-square infinite matrices exist?

Sorry, I tried to wrap my head more around this, but I failed. Given non-square matrix $A$ that has dimension $kn \times n$. Now let $n$ goto infinity. Is the matrix finally square?
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### Is it true that $|(Mv)\cdot(Nw) \leq C|v||w|$ (matrix-vector)?

If $M$ and $N$ are matrices such that each element of the matrices depends on $t$, so we have $M_{ij}(t)$, $N_{ij}(t)$, and we have the result $M_{ij}$, $N_{ij} \in L^\infty(0,T)$, is it true that ...
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### Linearly independent functionals

Let $f_1,\ldots,f_n$ be linearly independent linear functionals on a vector space $X$. Show that there are $n$ elements $x_1,\ldots,x_n$ in $X$ such that the $n\times n$ matrix $[f_i(x_j) ]$ is ...
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### question about a invertibility of a matrix

Let $Q_n$ a finite dimentional space of a Hilbert space $(Q,(\cdot,\cdot)_Q)$ and let $b$ a bounded bilinear form $b:Q_n\times Q_n\to \mathbb{R}$ such that it's elliptic, i.e., there exists $c>0$ ...
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### A matrix in $SL(2)$ has it's supremum norm and infimum fulfilled by orthogonal vectors

I am having trouble proving the next statement: If $B\in SL(2)$ and $||B||\neq 1$, for $||B||:= \underset{x\neq 0}{\sup}\big\{\frac{||B(x)||}{||x||} \big\}$, where $||\cdot||$ is the euclidian norm, ...
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### limit of evaluated automorphisms in a Banach algebra

Let $\mathcal{A}=\operatorname{M}_k(\mathbb{R})$ be the Banach algebra of $k\times k$ real matrices and let $(U_n)_{n\in\mathbb{N}}\subset\operatorname{GL}_k(\mathbb{R})$ be a sequence of invertible ...
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### Skew symmetric matrix decomposes

I am supposed to show that for a skew-symmetric matrix $A$ with $det(A) \neq 0$, meaning that is has an even number of columns and rows, there is an invertible matrix $R$ such that $R^T A R = M$, ...
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### Uniform convergence and interchange or sum and integral in Cauchy integral formula

I am working with the Cauchy Integral Formula for a matrix $A$ over a closed contour $C$. I have the following calculation, I believe this is correct, but I don't understand why I am allowed to ...
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### Hilbert norm and Euclidean distance

For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
When I read the proof of the Lie product formula in Reed Simon's book on functional analysis (which essentially reduces to showing $\left\Vert X_n - Y_n\right\Vert\rightarrow 0$ as $n\rightarrow 0$ ...
Suppose for every $x \in \mathbb{R}$ and $y \in [0,1]$, $M(x,y)$ is an $n$ by $n$ matrix and suppose that for every $y \in [0,1]$, $M(x,y) \to M_\infty$ as $|x| \to \infty$, where $M_\infty$ is a ...