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The Inverse and Implicit Function Theorems. Proposition. Suppose X and Y are normed vector spaces and L is a linear isomorphism from X onto Y .
 

Summary: The Inverse and Implicit Function Theorems.
Proposition. Suppose X and Y are normed vector spaces and L is a linear isomorphism from X onto Y .
Then
1
||L-1||
= inf{|L(x)| : x X and |x| = 1}.
Remark. In what follows 1/ = 0 and 1/ = 0.
Proof. Set = inf{|L(x)| : x X and |x| = 1}.
For any x X such that |x| = 1 we have
1 = |L-1
(L(x))| ||L||-1
|| |L(x)|
which implies that 1/||L-1
|| .
For any y Y we have that
|y| = |L(L-1
(y))| ||L-1
(y)|
which implies that ||L-1
|| 1/.

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics