Let and be as above, and let be a bounded operator. Then the norm of is defined as the real number
Now for any , if we let , then linearity implies that
and thus it easily follows that
To avoid cumbersome notational stuff usually one can simplify the symbols like and by writing only , since there is a little danger in confusing which is space about calculating norms.
0.0.1 TO DO:
Theorem Suppose is a linear map between normed vector spaces and . The following are equivalent:
Lemma Any bounded operator with a finite dimensional kernel and cokernel has a closed image.
Proof By Banach’s isomorphism theorem.
- 1 E. Kreyszig, Introductory Functional Analysis With Applications, John Wiley & Sons, 1978.
- 2 G. Bachman, L. Narici, Functional analysis, Academic Press, 1966.
|Date of creation||2013-03-22 14:02:17|
|Last modified on||2013-03-22 14:02:17|
|Last modified by||bwebste (988)|