# pde5 - 60 BRUCE K DRIVER 5 A very short introduction to...

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60 BRUCE K. DRIVER 5. A very short introduction to generalized functions Let U be an open subset of R n and (5.1) C c ( U )= K @@ U C ( K ) denote the set of smooth functions on U with compact support in U. De f nition 5.1. A sequence { φ k } k =1 D ( U ) converges to φ D ( U ) , i f there is a compact set K @@ U such that supp( φ k ) K for all k and φ k φ in C ( K ) . De f nition 5.2 (Distributions on U o R n ) . A generalized function T on U o R n is a continuous linear functional on D ( U ) , i.e. T : D ( U ) C is linear and lim n →∞ h T,φ k i =0 for all { φ k } D ( U ) such that φ k 0 in D ( U ) . Here we have written h T,φ i for T ( φ ) . We denote the space of generalized functions by D 0 ( U ) . Example 5.3. Here are a couple of examples of distributions. (1) For f L 1 loc ( U ) de f ne T f D 0 ( U ) by h T f i = R U φfdm for all φ D ( U ) . This is called the distribution associated to f.

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pde5 - 60 BRUCE K DRIVER 5 A very short introduction to...

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