4

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} \frac{|x-y|_k}{2^k(1+|x-y|_k)} $$ where $|.|_k$ is a separating family of semi-norms. My questions are the following :

1/ I have noticed that a sequence $(x_n)$ converges to a $x$ for $d$ if and only if for all $M\in \mathbb{N}$ $$\lim_{n\to \infty} \max_{k\leq M}|x_n-x|_k =0$$ However i did not find this result in textbook (probably because it is too easy or because it is wrong). Could someone tell me if it really holds of if i made a misstake ?

2/ The topology induced by the familly of semi-norms is the set of subsets of $V$ which are unions of sets of the form $$\cap_{j\in J}\{ y\in V : | |y|_k - a_j | <\epsilon_j \}$$ where $J\subset \mathbb{N}$ is finite. Is there a very short proof that this topology is the same as the one induced by the metric above ?

Alex Ravsky
  • 90,434
Guest0909
  • 105
  • About the first question. The maximum of a finite set of non-negative sequences tends to zero iff they all do. I wrote something about a similar question here, but those comments may not be very helpful to you, as the focus was different. – Jyrki Lahtonen Apr 18 '15 at 17:40
  • ??? No, it doesn't help me. Thank you. – Guest0909 Apr 18 '15 at 19:52

1 Answers1

1

1/ It seems the following.

Yes, both the necessity and the sufficiency can be easily checked. Let $\{x_n\}$ be a sequence of vectors of the space $V$.

Necessity. Assume that the sequence $\{x_n\}$ converges to a vector $x\in V$, $k\in\Bbb N$ be an arbitrary index, and $\varepsilon>0$ be arbitrary. Choose a number $\delta>0$ such that $\frac{2^k\delta}{1-2^k\delta}<\varepsilon$. Since the sequence $\{x_n\}$ converges to a vector $x$, there exists a number $N$ such that $d(x_n,x)<\delta $ for each index $n>N$. Let $n>N$ be an arbitrary index. Then

$$\frac{|x_n-x|_k}{2^k(1+|x_n-x|_k)}\le d(x_n,x)<\delta,$$

so $$|x_n-x|_k<\frac{2^k\delta}{1-2^k\delta}<\varepsilon. $$

Sufficiency. Assume that there exists a vector $x\in V$ such that for each index $k\in\Bbb N$ a sequence $\{|x_n-x|_k\}$ converges to the zero. Let $\varepsilon>0$ be an arbitrary number. Choose a natural number $M$ such that $\frac 1{2^M}<\frac{\varepsilon}{2}$. Since for each index $k\le M$ the sequence $\{|x_n-x|_k\}$ converges to the zero, there exists a number $N$ such that $|x_n-x|_k<\varepsilon/2$ for each index $k\le M$ and each index $n>N$. Let $n>N$ be an arbitrary index. Then

$$d(x_n,x) = \sum_{k\in \mathbb{N}} \frac{|x_n-x|_k}{2^k(1+|x_n-x|_k)} =$$ $$\sum_{k=1}^M \frac{|x_n-x|_k}{2^k(1+|x_n-x|_k)} + \sum_{k=M+1}^\infty \frac{|x_n-x|_k}{2^k(1+|x_n-x|_k)}<$$ $$\sum_{k=1}^M \frac{\varepsilon}{2^{k+1}} + \sum_{k=M+1}^\infty \frac{1}{2^k}<$$ $$\frac{\varepsilon}2+\frac 1{2^M}<\frac{\varepsilon}2+\frac{\varepsilon}2<\varepsilon.$$

man_in_green_shirt
  • 2,598
Alex Ravsky
  • 90,434
  • Is there a good reference for Fréchet spaces, and more generally Fréchet manifolds, and their topology? – QGravity Apr 07 '18 at 03:03