# 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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### Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in $E^{**}$...
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### Differentiation operator on smooth function with compact support

Suppose $f$ is $C^\infty$ with compact support. Let $T_n$ be the operator which sends $f$ to its $n$-th derivative. Is $||T_nf||_\infty$ bounded? It seems like I should use Stone-Weierstrass, but ...
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### norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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### Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
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### On an inequality concerning vectors with norm less than one

Consider two sets of vectors $\{v_i\}_{i=1}^{n_1}$ and $\{w_j\}_{j=1}^{n_2}$ with $v_i,w_j\in\mathbb{R}^n$ such that $\|v_i\|_2< 1$ and $\|w_j\|_2< 1$ for all $i=1,\dots,n_1$ and $j=1,\dots,n_2$....
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### Matrix with roots of unity entries

For a given prime p, i am interested in the norms of matrices which have root of unity entries, i.e., $M_{k,l} \in \{1, \zeta, \dots, \zeta^{p-1}\}$ where $\zeta = \exp{(2\pi I/p)}$. Are there any ...
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### Exponentially weighted function in $\mathcal L_1$?

I have an interesting adaptive control problem. Consider a signal $u(t)$ generated by normalizing another signal, so that $$0 \leq u(t) < 1.$$ Consider the function generated from $u(t)$ as ...
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### Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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### Operator norm of the inverse

If I made no mistake, one can calculate the operator norm of the inverse of any given (invertible) operator $A: V\rightarrow V$ via: \begin{align}\|A^{-1}\| & = \sup\left\{\frac{\|A^{-1}b\|}{\|b\|...
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### Show that usual $L^2$ norm is equivalent to arbitrary norm $||\cdot||$ that satisfies 'convergence condition'
Given $L^2(\mathbb{R})$ consider a norm $||\cdot||$ on $L^2$ such that $(L^2,||\cdot||)$ is a Banach space and every $||\cdot||$-convergent sequence has a subsequence that converges almost everywhere. ...