# 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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### If $A$ is bounded and dissipative, is $\lambda \mathbb 1-A$ invertible for $\lambda>0$?

Let $X$ be a Banach space and $j$ a map (not necessarily linear or anti-linear!) from $X$ to $X'$ so that $$j_x(x)=\|x\|^2 \qquad \forall x \in X \text{ and }\qquad \|j_x\|_{X'}=\|x\|$$ We say that ...
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### Entrywise expression for L2 matrix norm

The matrix norm induced by the $\ell^2$ norm is known to be equal to the maximum singular value of the matrix. The matrix norms induced by the $\ell^1$ and $\ell^\infty$ norms admit simple ...
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### Show that there is a bounded linear functional $\ell : \mathscr C [0,1]\to\mathbb R$ with $\lVert \ell \rVert \leq 1,\ \ell(1)=0,\ \ell(\cos(x))=1$.

The title says it all. I've been assuming that the best way to do this is constructively, by finding such an $\ell$. I have by a theorem in our class that since $\lVert \cos(x)\rVert =1$, we know that ...
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### Why is the following scaling good for the general case?

In the following post how come scaling to $x_0=0$ and $r=1$ can help to prove for any $r,x_0$? Is there a general rule for when is it OK to scale?
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### Generalized Poincaré Inequality on H1 proof.

let's see if someone can help me with this proof. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. And let $L^2\left(\Omega\right)$ be the space of equivalence classes of square integrable ...
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### Finding the closest element to a function in a normed space containing functions.

Let $B=\{f\in c[1,0]|\forall 0\leq x\leq 1 : f(x) \geq 0\}$. Given $f\in c[0,1]$, find the closest element in $B$ under the $\|\cdot\|_2$ norm. I can see something similar to this question in the ...
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I'd ask someone to edit my question, I'm not sure if it's correctly spelled. Let $X$ be a normed vector space with Hamel basis $\{e_n\}$,$n\in \mathbb{N}$ consistent in unit vectors $e_n$ and $Te_n=... 1answer 22 views ### Length of a curve under a non-Euclidean norm in the integral form. Let$V$be a normed space. Let$\gamma\colon [a,b] \rightarrow V$be continuous. Then$\gamma$is a curve. Let$P$be a partition of$[a,b]$, then $$\Lambda(\gamma, P) := \sum_{i=1}^n \| \gamma(x_i) -... 1answer 17 views ### Which one is the correct definition of natural norm? In the definition 2 of Normal Subgroup Reconstruction and Quantum Computation Using Group Representations, the authors have defined the natural norm of a matrix as follows. The natural norm of the ... 1answer 32 views ### Why the separate notation for norm One usually denotes the norm as \|\cdot\| , \| v\| := \sqrt{\langle v, v \rangle}. However, in metric spaces, one often writes d(x,y) \equiv \lvert x-y \rvert. Since the norm canonically ... 1answer 35 views ### Compactness in a vector space If E is a normed space and F is a subspace of E, how to prove that if F\neq\{0\} then F is not compact? I begin by this let x\in F then F=\bigcup_{x\in F} B(x,\varepsilon) how to say ... 0answers 16 views ### For an open subset D of a normed space, its multiple \alpha D is also open D is a subset of \mathbb R and E=\{\alpha x : x\in D, \alpha>0\} Prove that E is open iff D is open For each \alpha x \in E \exists a ball B_\epsilon (\alpha x)\subset E. Can ... 1answer 50 views ### Prove that this transformation inverse exists and it's bounded If X,Y are Normed Vectorial Spaces, T is a bounded lineal transformation. Prove that if exists b>0 such that \|Tx\|\geq b\|x\| \forall x\in X. Then T^{-1}:Y\rightarrow X exists and it's ... 1answer 10 views ### Proof about converging absolutely with respect to equivalent norm If series converges absolutely with respect to some norm, then it also converges absolutely with respect to any kind of equivalent norm. I need to prove this assertion, but I have no idea from where ... 0answers 10 views ### Minimizing matrix norm via left-multiplication by SL(m) Suppose that M is an m\times n matrix of full row rank, with m \leq n. Then if \|M\| is the matrix norm induced on M from the norm on our vector space, we can look for the following ... 0answers 19 views ### Is this question well-formed? Let T\in\mathcal B(X,Y),\ \lVert T\rVert<1,\ Y Banach, show \sum_{n=0}^\infty T^n\in\mathcal B(X,Y). If X=Y, then I think I have solved this problem entirely already. But if X\neq Y, then I don't understand what is being asked. How is T^n defined when X\neq Y? When I consider X=Y, then I ... 1answer 28 views ### prove the map is a contraction Let L be a fixed positive real number. Let K:[0,T] \times \mathbb{R} \rightarrow \mathbb{R} be continuous and satisfy the lipschitz condition |K(s,x)-K(s,y)| \leq L|x-y| for all s \in [0,T]... 1answer 30 views ### Why is \ker(T-\lambda I)^n finite-dimensional? Let X be a normed space and T be a compact operator on X and \lambda \in \sigma(T)\setminus\{0\}. A closed unit ball in \ker(T-\lambda I) always admit a convergent subsequence of a sequence,... 1answer 30 views ### Closed convex hull = closure of convex hull? If the "closed convex hull" of A is the intersection of all closed convex sets containing A, is this the same as the closure of the convex hull of A? Many have asked whether the closure of the convex ... 0answers 18 views ### Find \langle f,g \rangle w.r.t. L_0 \perp L_1. Let X=C[-1,1], and L_k= \{ <t^{k+2i}, i=0,1,2,... > \} . Define an inner product on X with respect to L_0 \perp L_1. Then confirm that L_0 \perp L_1 on your inner product. Can we ... 1answer 18 views ### some detail calculation on the proof of equivalence of norms We say that two norm \|x\|_1 and \|x\|_2 on a vector space X are said to be equivalent if there exists K>0 and M>0 such that$$ K\|x\|_1\le \|x\|_2\le M\|x\|_1 $$Prove that on a ... 1answer 53 views ### Find the norm of a linear continuous operator in X=C([0,1],\mathbb{R}) with the norm \|f\|_2=\sqrt{\int_0^1 f^2(x)dx} we define T:X\rightarrow X by Tf=gf for g\in X How to prove that T is continuous and how to find \|T\| ? I find ... 2answers 27 views ### Is T(\ell^1 ) \subseteq \ell^1? If we have a linear operator (\ell^{\infty} , \|\cdot \|_{\infty} ) \rightarrow (\ell^{\infty} , \|\cdot \|_{\infty} ) by T((a_k)_{k \ge 1}) = (b_k)_{k \ge 1} where$$ b_k= \frac{a_1+...+a_k}{k}... 1answer 47 views ### Cauchy but not rapidly Cauchy I want to show that the sequence$\{\frac{(-1)^n}{n}\}$is Cauchy but not rapidly Cauchy. Here is the work I done so far. I am curiously if I made any errors. Consider the normed linear space$\...
Let $X$ be a normed vector space over $\mathbb{K}$. Then there is only one completion of $X$, the banach space $\hat{X}$ such that $X$ is a dense subspace of $\hat{X}$. I am trying to prove that there ...