b****t
发帖数: 114 | 1 A closed set A \in R^n and a compact set B \in R^n
show that A + B is closed, where A + B = {z: z=x+y, x \in A, y \in B, and z
\in R^n}.
provide a example that if B is closed but not compact set, then the result
does not hold.
谢谢!
beet | B****n
发帖数: 11290 | 2 1. Assume z_n\in A+B and z_n->z,
there exists x_n\in A, y_n\in B such that z_n=x_n+y_n.
Since B is compact, there exists a subsequence of y_n' such that y_n'->y\in
B
so there exists a subsequence x_n'->x\in A
so z=x+y\in A+B
2. choose A={1,2,3,...} B={-1,-2+1/2.-3+1/3,...}
z
【在 b****t 的大作中提到】
: A closed set A \in R^n and a compact set B \in R^n
: show that A + B is closed, where A + B = {z: z=x+y, x \in A, y \in B, and z
: \in R^n}.
: provide a example that if B is closed but not compact set, then the result
: does not hold.
: 谢谢!
: beet
| u*****n
发帖数: 28 | 3
in
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~这是为啥？
~~~~~~~~~~~~~~~~这个是怎么得出来的？
【在 B****n 的大作中提到】
: 1. Assume z_n\in A+B and z_n->z,
: there exists x_n\in A, y_n\in B such that z_n=x_n+y_n.
: Since B is compact, there exists a subsequence of y_n' such that y_n'->y\in
: B
: so there exists a subsequence x_n'->x\in A
: so z=x+y\in A+B
: 2. choose A={1,2,3,...} B={-1,-2+1/2.-3+1/3,...}
:
: z
| B****n
发帖数: 11290 | 4 x_n'=z_n'-y_n'->z-y. Because A is closed z-y is in A
Define x=z-y, so z=x+y is in A+B
【在 u*****n 的大作中提到】
:
: in
: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~这是为啥？
: ~~~~~~~~~~~~~~~~这个是怎么得出来的？
|
|
