Chủ đề này có 2 trả lời

[hoc.toan]

Problem 1:

Let $\{x_n\}\subset (X,d)$ satisfying $d(x_{n},x_{n-1})< \varepsilon, n--->+\infty,\forall \varepsilon >0, $.

$\{x_n\}$ is a Cauchy sequence? Give some encounter examples? If not, in which condition $\{x_n\}$ is a Cauchy one?

Bài viết đã được chỉnh sửa nội dung bởi hoc.toan: 24-06-2007 - 09:01

[hoc.toan]

Problem 2:

Let $L(X,Y)$ be space of all linear operators between two normed spases X,Y.

Check this

$inf\{M>0,\|Tx\|\leq M\|x | }=\sup_{\|x\|\leq 1}\|Tx\|=\sup_{x\neq 0}\dfrac{\|Tx\|}{\|x\|}= \sup_{|x|= 1}|Tx|, \forall T\in L(X,Y)$

in the following cases:

2.1. T is continuous.

2.2. T is semi-continuous.

2.4. T is discontinuous.

If one doesn't hold, please give some encounter examples.

Thanks.

Bài viết đã được chỉnh sửa nội dung bởi hoc.toan: 24-06-2007 - 09:10

[thang ngo]

try to type by E.
the answer is the x_{n} isn't cauchy sequence, here is a counterexample :
x_{n}=1/n , that's ok.