bounded operator
Definition [1]
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1.
Suppose and are normed vector spaces with norms and . Further, suppose is a linear map . If there is a such that for all we have
then is a bounded operator.
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2.
Let and be as above, and let be a bounded operator. Then the norm of is defined as the real number
Thus the operator norm is the smallest constant such that
Now for any , if we let , then linearity implies that
and thus it easily follows that
In the special case when is the zero vector space, any linear map is the zero map since . In this case, we define .
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3.
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:
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1.
The defined norm for mappings is a norm
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2.
Examples: identity operator, zero operator: see [1].
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3.
Give alternative expressions for norm of .
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4.
Discuss boundedness and continuity
Theorem [1, 2] Suppose is a linear map between normed vector spaces and . If is finite-dimensional, then is bounded.
Theorem Suppose is a linear map between normed vector spaces and . The following are equivalent:
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1.
is continuous in some point
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2.
is uniformly continuous in
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3.
is bounded
Lemma Any bounded operator with a finite dimensional kernel and cokernel has a closed image.
Proof By Banach’s isomorphism theorem.
References
- 1 E. Kreyszig, Introductory Functional Analysis With Applications, John Wiley & Sons, 1978.
- 2 G. Bachman, L. Narici, Functional analysis, Academic Press, 1966.
Title | bounded operator |
---|---|
Canonical name | BoundedOperator |
Date of creation | 2013-03-22 14:02:17 |
Last modified on | 2013-03-22 14:02:17 |
Owner | bwebste (988) |
Last modified by | bwebste (988) |
Numerical id | 19 |
Author | bwebste (988) |
Entry type | Definition |
Classification | msc 46B99 |
Related topic | VectorNorm |