# Tagged Questions

Norm is a function on a vector space $X$ which generalizes notion of length of vector in general vector spaces.

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### Sequence of functions $\sin^{n} (\pi x)$ cauchy in C[0,1]

Equipping C[0,1] with its usual (supremum) norm, for each $n \in \mathbb{N}$, let $g_n(x)=\sin^{n} (\pi x)$ for $x \in [0,1]$. I'm trying to prove (or disprove) this sequence is cauchy. Intuitively, ...
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### Condition number of 2x2 nonsingular matrix

I am working on the problem that I have to show why the infinity-norm condition number and 1-norm condition number of 2x2 nonsingular matrix are equal. MY ATTEMPT: Since $I=AA^{-1}$, the condition ...
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### Write the norm $||R^h v||$ just in function of $v$ if $R=[ Q \, \mathbf{0}]$ and the columns of $Q$ form an orthonormal basis

Let $Q$ be an $m \times (m-k)$ (complex) matrix where its columns form an orthonormal basis ($m-k$ vectors). We define matrix $R=[ Q \, \, \mathbf{0}]$, where $\mathbf{0}$ is the $m \times k$ zero ...
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### Source/explanation for this matrix inequality

Here it is: $$z^\top M^{-1} M^{-1}z \le \|M^{-1}\| z^\top M^{-1} z.$$ Where $\pmb M$ is positive definite symmetric, $z$ is a vector in $\mathbb{R}^p$ (not necessarily normed!) and $||\pmb M||$ is ...
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### Open mapping theorem for normed abelian groups

A norm on an abelian group is a function valued in $\mathbb{R}_{\geq 0}$ which satisfies $|x|=0 \Leftrightarrow x=0$, $|{-}x|=|x|$, and $|x+y| \leq |x|+|y|$, not necessarily $|z x| = |z| |x|$ for ...
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### Exchanging limits with norms and linear functionals

In a normed vector space $X$, when can we say: $\lim\|x_n\|=\|\lim x_n\|$ and further, if $f\in X^{*}$, when can we say: $\lim fx_n=f(\lim x_n)$?
Suppose we have a symmetric real matrix $\pmb M$ satisfying: $$\underset{\pmb\alpha\in\mathbb{R}^p:||\alpha||=1}{\min}\;\pmb \alpha\pmb M\pmb \alpha^{\top}\geqslant k>0.$$ Then, I am trying to ...
Stumbled upon this one in a textbook: Let there be a linear transformation $T:V\rightarrow V$ over a finite inner product space $V$. It is known that $TT^* = 7T - 12I$. How can it be proved that ...