# Tagged Questions

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

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### Is it possible to characterize completeness of a normed vector space by convergence of Neumann series?

If $X$ is a normed vector space and if for each bounded operator $T \in B(X)$ with $\| T\| < 1$, the operator ${\rm id} - T$ is boundedly invertible, does it follow that $X$ is complete? Context: ...
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### completeness of $l_1 ^\infty$

I'm trying to prove that $l_p ^\infty$ is complete for each $p\geq 1$ but only with the definition of $\varepsilon$-$N$. I know that this have been proved in other posts here but I couldn't find a ...
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### When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
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### Operator Norm and Submultiplicativity against the Spectral Norm

Consider $\mathcal{A}:\mathbb{R}^{n\times m}\to \mathbb{R}^{p\times q}$ to be a linear operator. I know that by considering the trace norm and using the submultiplicativity of the operator norm we ...
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### Find all $p \ge 1$ for which the Hölder norm $\|\cdot\|_p$ is generated by a scalar product.

Find all $p \ge 1$for which the Hölder norm $$\|x\|_p := \left(\sum^{n}_{i=1} |x_i|^p\right)^{\frac{1}{p}}$$ is generated by a scalar product. We know that norm is generated by a scalar product ...
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### Gelfand width of $l_1$ unit ball in $\infty$ metric

The $l_1$ balls in $l^N_p$ are well studied, but I cannot find anything about estimates of $d^m(B^N_1,l^N_\infty)$ except of $d^m(B^N_1,l^N_\infty)\geq C\min\{1,\frac{\ln(eN/m)}{m}\}$ which is not ...
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### Finding L^1 centers of sets of probability distributions

Let $\mathcal{P}^n = \{ x \in \mathbb{R}^n : x \geq 0, \sum x = 1\}$. Suppose I have $p_1, \ldots, p_m \in \mathcal{P}^n$. I want to find an $L^1$ center for these points. i.e. $q \in \mathcal{P}^n$ ...
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### Do the $p$-norms in $\mathbb R^2$ have a nice geometric intuition behind them?

For $p\geq 1,$ the $p$-norm of a vector $(x,y)\in\Bbb R^2$ is the number $\|(x,y)\|_p=(|x|^p+|y|^p)^{1/p}.$ I learned this definition some time ago, but I never really understood it. Is there a ...
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### linear problem with $\|.\|_\infty$ and $\|.\|_1$ norm constraints

I have a question regarding a straightforward linear algebra problem, yet the solution is (at least for me) not trivial. Assume the sequences $\phi_i$ with coefficients $\phi_i[n]\in\mathbb{R}$, and ...
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### Name for the universal normed space associated to a seminormed space

If $(V,p)$ is a seminormed space, then $(V/N,\overline{p})$ is a normed space, where $N=\{x \in V : p(x)=0\}$ and $\overline{p}(x \bmod N) = p(x)$. My question is as follows: Is there a common name ...
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