| Line 9: | Line 9: | ||
<math>(z,w)\mapsto z+w</math> | <math>(z,w)\mapsto z+w</math> | ||
| − | is a continuous mapping | + | is a continuous mapping |
| + | |||
| + | <math>(\mathbb C\times \mathbb C)\to\mathbb C</math> | ||
| + | |||
| + | because | ||
<math>|(z+w)-(z_0+w_0)|\le|z-z_0|+|w-w_0|</math> | <math>|(z+w)-(z_0+w_0)|\le|z-z_0|+|w-w_0|</math> | ||
| − | and to make this last quantity less than <math>\epsilon</math> | + | and to make this last quantity less than epsilon, it suffices to take |
| + | |||
| + | <math>|z-z_0|<\epsilon/2</math> | ||
| + | |||
| + | and | ||
| − | + | <math>|w-w_0|<\epsilon/2.</math> | |
To handle complex multiplication, you will need to use the standard trick: | To handle complex multiplication, you will need to use the standard trick: | ||
<math>zw-z_0w_0 = zw-zw_0+zw_0-z_0w_0=z(w-w_0)+w_0(z-z_0)</math>. | <math>zw-z_0w_0 = zw-zw_0+zw_0-z_0w_0=z(w-w_0)+w_0(z-z_0)</math>. | ||
Revision as of 07:35, 3 September 2009
Homework 2
Here's a hint on I.8.3 --Bell
It is straightforward to show that
$ (z,w)\mapsto z+w $
is a continuous mapping
$ (\mathbb C\times \mathbb C)\to\mathbb C $
because
$ |(z+w)-(z_0+w_0)|\le|z-z_0|+|w-w_0| $
and to make this last quantity less than epsilon, it suffices to take
$ |z-z_0|<\epsilon/2 $
and
$ |w-w_0|<\epsilon/2. $
To handle complex multiplication, you will need to use the standard trick:
$ zw-z_0w_0 = zw-zw_0+zw_0-z_0w_0=z(w-w_0)+w_0(z-z_0) $.
