# Tagged Questions

35 views

### Limit of nth power of operator norm

I am given a compact operator $A$ which lives in a Banach algebra and whose spectral radius obeys $\rho(A)=\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}<1$. Now I want to prove that this implies ...
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### what is the limit of $l_p$ at p=0?

The p-norm is defined as: $$\ \|x\|_p=\left(|x_1|^p+|x_2|^p+\dotsb+|x_n|^p\right)^{\frac{1}{p}}$$ When $p<1$, this is no longer a "norm" because it violates the triangle inequality (- it is ...
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### Geometric Mean limit of $\ell_p$ norm of sums

My analysis professor introduced the $\ell_p$ norm to our class as: \begin{align} \| x \|_p = \left(\frac{1}{n}\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} We are asked to prove the following: ...
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### Example of a sequence that converges to two different limits with respect to two complete norms

I've wondered about the following question : Is there an (explicit?) example of a vector space $X$, two complete norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on $X$, and a sequence $(x_n) \subseteq X$ such ...
Can a norm "grow exponentially"? Let $||\cdot||_*: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$ be a norm such that: $$\lim_{|x| \rightarrow \infty } \frac{ ||x||_* }{ e^{|x|} } > 0$$ where ...
### Convergence of $\frac1m(I+A+A^2+\cdots+A^{m-1})$
Let $A$ be an $n\times n$ matrix of nonnegative entries such that $A_{i1}+A_{i2}+\cdots+A_{in}=1$ for all $i\in\{1,2,\ldots,n\}$. What does $A$ have to satisfy so that the sequence ...