# 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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### $L^p$ and $L^1$ norms

As $L^p$ norm of $f:\mathbb R\to \mathbb R^{n}$ is defined as $$\|f(t)\|_p:=\left(\int_S\|f(t)\|^pdt\right)^{1/p}$$ I have two questions: What kind of norms for the function $f(t)$ in the ...
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### Is this a property of Normed Vector Spaces?

Let $X$ a normed vector space on $(\mathbb{K}, + , . )$. Is the following assertion true? Any $x$ of $X$ can be written as $x = \alpha a$ , $\alpha \in \mathbb{K}$ , $a \in X$ with $||a||_X=1$ ...
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### Are the infinity norm and the the induced norm equivalent in $l^2$?

Consider the function $||\cdot||_\infty:l^2\to[0,\infty)$ given by $$||(a_n)||_\infty = \sup\{|a_n|:n\ge 1\}.$$ So I want to know if this norm is equivalent to: $$||(a_n)||=\sqrt{\sum |a_n|^2}$$ ...
1answer
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### Open Ball and Lipschitz Equivalence equivalence

I am trying to show that two norms $\|\cdot\|$ and $\|\cdot\|^\prime$ are Lipschitz equivalent if and only if there exist numbers $r,R >0$ such that $B_r \subseteq B_1^\prime \subseteq B_R$ where ...
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### Lower bound on multiplicative norm of Banach algebra.

Let $A$ be a Banach algebra over $\mathbb{C}$ and $N: A \to \mathbb{R}_{\ge 0}$ the corresponding multiplicative norm. For $a \in A$, do we have$$\limsup_{n \to \infty} (N(a^n))^{1\over{n}} \le N(a)?$$...
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### Geometry of a Cauchy sequence in a normed space

A sequence in a normed space $X$ is called a Cauchy sequence if and only if for every $\epsilon > 0$ there exists an integer $N\in \Bbb N$, such that $\|x_n-x_m\|\lt \epsilon$ for all $n>m>N$....
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### Deducing equivalence between norms from simple condition

Let $||\cdot||,\;||\cdot||'$ be norms on $V$. Suppose for some $a,b>0$ we have: $$||x||<a\Rightarrow ||x||'<1\Rightarrow ||x||<b$$ Show that $||\cdot||,\;||\cdot||'$ are Lipschitz ...
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### How can I prove this proposition of linear algebra?

Good afternoon! I have to show this proposition: 1) Let $A \in \mathbb R^{n\times n}$ a non singular matrix and $PA=LU$, $P$ permutation matrix, $L$ a lower triangular matrix with $1$ on its ...
1answer
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### “Decay” of $L^2$-norm of the solution of heat equation with mixed boundary conditions

I'm considering the heat equation $$u_t = u_{xx}$$ on the interval $[0, \pi]$ with mixed boundary conditions $$u(0,t)=0 \quad \text{and} \quad u_x (\pi,t)=0,$$and smooth initial data $u(x,t)=u_0 (x)$ ...
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### Prove by Induction that Norms in a Finite Dimensional Space are Equivalent?

I would have thought that this is a good candidate for an inductive proof, but I have searched for one and failed. Is there such a proof, and if not why not ? Here's how far I got. It's easy to ...
1answer
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### $p$-norms inequalities [closed]

I got this math homework and I can't do it. I have to prove the inequalities for p-norms in $\mathbb R^n$: $\|x\|_1\ge\|x\|_2\ge\|x\|_\infty$ $\|x\|_1\le\sqrt{n}\|x\|_2$ (I have already proven this ...
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### soft thresholding derivation with two terms

I am trying to minimize the following function $$argmin_x||x-y||^2_2 + ||x-z||_2^2 + \lambda || \frac{x- w}{c}||_1$$ I have been doing it for while but i am not sure how to do it. I have already been ...