# Questions tagged [norms]

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### Estimate of Hölder Norms (Littlewood–Paley theory)

I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem: Recall that ...
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### Green's identity with a different norm

Let $\Omega \subset \mathbb{R}^n$ be a domain with a smooth boundary $\Gamma$. Suppose that $f, g \colon \mathbb{R}^n \to \mathbb{R}$ are of class $C^\infty( \overline{\Omega})$. Then Green's first ...
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### Bounding supremum norm of Lipschitz function by L1 norm

Consider $f:[0,1]^d \to \mathbb{R}$. Suppose that $f$ is $L$-Lipschitz w.r.t. the Euclidean norm. Can we provide an upper bound on $\|f\|_\infty$ in terms of $\|f\|_1 := \int_{[0,1]^d} |f(x)|dx$ ? In ...
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### Double dual of an extended seminorm

I have already ask this question on MSE but it didn't received any answer during the weekend. So I thought I will ask it here too. Please let me know if it is bad practice and/or which of MO and MSE ...
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### Subtracting the weak limit reduces the norm in the limit

Question Let $X$ be some reflexive Banach space. Suppose $x_n$ is some sequence in $X$ that weak converges to some $y \neq 0$. Is it the case that $$\limsup \|x_n - y\| < \limsup \|x_n\| ?$$ ...
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### Ratio of maximum to minimum value

Let $y = X \beta + \epsilon$, where $y \in R^{n}$, $X \in R^{n \times p}$, $\beta \in R^{p}$ and $\epsilon \in R^{n}$. Let $X = USV^\top$ be the SVD of the $X$. Let $u_i$ be the rows of $U$, then ...
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### How to minimize l1-norm constrained by “infinity norm”

Let $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^m$. I have the following two problems: P.1. \begin{equation} \underset{x\in\mathbb{R}^n}{\text{minimize}} \| Ax-b \|_1 \\ \text{s.t. } \| x \...
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### Is this an error in Loomis and Sternberg?

In Loomis and Sternberg's Advanced Calculus section 3.3 Continuity, they make this comment (just before Theorem 3.2, [pp. 128-129 in my copy): A linear map $T : V \rightarrow W$ is bounded below by ...
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### Cartesian product of Banach spaces: all norms such that the inclusion is an isometry are equivalent?

Let $\mathcal{A}$ be an arbitrary (typically infinite-dimensional) Banach space with norm $\|\cdot\|_{\mathcal{A}}$ and let $\mathcal{A}^{n}$ be its Cartesian product. I came across the following ...
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### $L^p$ norm inequalities with respect to strongly-log-concave densities

Let $\pi(x)=\frac{e^{-f(x)}}{\int_{\mathbb{R}^d}e^{-f(u)}du}$ be a strongly-log-concave distribution, i.e., $f(x):\mathbb{R}^d\rightarrow R$ is an $m$-strongly convex function. Also, $f(x)$ has $L$-...
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### When does the map from a normed vector cone to its double dual preserve norms?

If $V$ is a normed vector space then the natural map from $V$ to its double dual $V''$ is norm-preserving as follows from Hahn-Banach theorem. This is well-known. Now assume that P is just a vector ...
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### Is the union of l^p a Banach space under some norm?

As a set of sequences, take the union of $\ell^p$, $p\geq 1$. As $p$ increases, the $\ell ^p$ space is larger, with strict inclusion. However, this infinite union is strictly contained in $c_0$, ...
Question: What is the state of the art on analytic approximations of $\min$ and $\max$? My hunch is that numerical analysts probably have a better solution than the one I propose here. For any $\... 0answers 148 views ### Spectral norm of difference of quadratic matrices restricted to a subspace Say that we have two matrices$X$and$Y$of dimensions$(T \times N)$with$N < T$and$rank(X)=rank(Y)=N$. Furthermore, define a$(T \times k)$dimensional matrix$D$with$k<N$and$rank(D)=k$... 1answer 79 views ### Sequence converging to different limits with respect to two different _complete_ norms Do there exist a real vector space$X$complete with respect to norms$|\cdot|$and$\|\cdot\|$and a sequence$(x_n)_{n\in \mathbb N} \subset X$such that there exist$x,y\in X$:$x\ne y$,$|x_n - x|\...
If I have a nonsymmetric matrix whose operator norm is $\leq 1$ and square root it, does its operator norm remain below $1$? More formally, I want to know whether there is always at least one square ...