1.3 Continuous mapping and function
Definition 1.3.1
A mapping
Note When
Theorem 1.3.3
If
Proof
For any
When
Definition 1.3.4
A set
Theorem 1.3.5
If
Example 1.3.1 Continuous mappings and functions.
- Constant mapping.
-
Linear mapping
. For any ,So
is a Lipschitz function, is uniformly continuous. - Norm
. , Lipschitz. - The addition
, number multiplication and inner product between vectors. are continuous, then where is also continuous. is continuous and , then where is also continuous.
Example 1.3.2 If arbitrarily fixing one or more variable(s) will get a continuous function, does it mean the original function is continuous? No! See the example below.
Example 1.3.3 Is
Fix
Now consider the continuity of binary
Now consider
Another example is
Example 1.3.4 Application. Assuming a linear mapping
Proof
Let
It is to be proved that
Then the spectral decomposition is proved by mathematical induction.