# Questions tagged [norms]

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### inequivalent norms [closed]

I am thinking about the following question: Let $X$ be a Banach space, say separable, e.g., $l_p$ or $c_0$. When can I say that there exist inequivalent complete norms on $X$?
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### Holder inequality for a general rectangular matrix

Let $A \in \mathbb{R}^{m\times n}$ and $p,q \in \mathbb{R}^{+}$ such that $\frac{1}{p}+\frac{1}{q}=1$. I am interested to prove the following: $$\|A\|_{p}=\|A^T\|_q$$ I have tried using Holder ...
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### Relation between Frobenius, spectral norm and sum of maxima

Let $A$ be a $n \times n$ matrix so that the Frobenius norm squared $\|A\|_F^2$ is $\Theta(n)$, the spectral norm squared $\|A\|_2^2=1$. Is it true that $\sum_{i=1}^n\max_{1\leq j\leq n} |A_{ij}|^2$ ...
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I have the following question concerning an estimate of the total variation norm. Let $\mu$ be a bounded Borel measure on $\mathbb{R}$ and denote by $\mu_t$ the measure defined by $\mu_t(\Omega):=\mu(\... 1answer 227 views ### Does this norm have a specific name? Banach space? References? Let$(X,\mathscr{B},\mu)$be a$\sigma$-finite measure space. Let$\gamma$be a probability measure on$L_2(\mu)$with$\mathrm{supp} \, \gamma = L_2(\mu)$and existing first moment. Then $$f \... 0answers 242 views ### Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1) We need a help to find a reasonable condition such that the spectral radius for a special matrix \mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}} is smaller ... 1answer 213 views ### Largest eigenvalue of product of orthogonal-projection rank-1 perturbation Suppose I have a symmetric positive definite matrix A \in \mathbb{R}^{n \times n} with n linearly indepedent columns a_1,...a_n in \mathbb{R}^n. All columns a_i has norm 1, but they are not ... 2answers 279 views ### Existence of p=-infinity norm [closed] Given a vector \mathbf {x} =(x_{1},\ldots ,x_{n}) the p-norm is defined as$$ \left\|\mathbf {x} \right\|_{p}:={\bigg (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\bigg )}^{1/p}$$for$p \geq 1$. For$...
The following question looks simple, but the answer is not obvious for me: Let $S$ be a $*$-algebra and $\left\Vert \cdot \right\Vert _{1}$, $\left\Vert \cdot \right\Vert _{2}$ $C^*$-norms on $S$ ...
Let $F$ be a continuously differenable function over $\mathbb{R}$. Let $\Omega$ be a bounded subset of $\mathbb{R}^2$. Assume that for every $w\in L^2(\Omega)$ then $v(x)=F(w(x))$, $x\in \Omega$, is ...