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Unformatted text preview: 1 Lecture 10 Recall, a problem is said to be wellposed if 1. there exists a solution 2. the solution is unique 3. the solution depends continuously on the data. In terms of an operator equation Aφ = f where A : U → V, U, V subsets of normed space X and Y , respectively, this means A : U → V is bijective and A 1 : V → U is continuous. So there are 3 types of illposedness 1. If A is not surjective (onto), then Aφ = f is not solvable for all f ∈ V (nonexistence of a solution). 2. If A is not injective (one to one), then Aφ = f may have more than one solution (nonuniqueness). 3. If A 1 exists, but is not continuous, then the solution does not depend continuously on the data (instability). These properties are not completely independent. Theorem 1 If A : X → Y is a bounded linear operator mapping a Banach space X bijectively into a Banach space Y , then A 1 : Y → X is bounded and therefore continuous....
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 Fall '08
 Staff
 Vector Space, Sets, Hilbert space, kN, Compact operator, Bounded operator

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