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Unformatted text preview: MATH 220: DISTRIBUTIONS AND WEAK DERIVATIVES ANDRAS VASY Suppose V is a vector space over F = R or F = C . The algebraic dual of V is the vector space L ( V, F ) consisting of linear functionals from V to F . That is elements of f ∈ L ( V, F ) are linear maps f : V → F satisfying f ( v + w ) = f ( v ) + f ( w ) , f ( cv ) = cf ( v ) , v,w ∈ V, c ∈ F . When V is infinite dimensional, we need additional information, namely con tinuity. So if V is a topological space with the compatible with the vector space structure, i.e. if V is a topological vector space, we define the dual space V ∗ as the space of continuous linear maps f : V → F . For us, V is the class of ‘very nice objects’, and V ∗ will be the class of ‘bad objects’. Of course, normally there is no way of comparing elements of V with those of V ∗ , so we will also need an injection ι : V → V ∗ so that elements of V can be regarded as elements of V ∗ (by identifying v ∈ V with ι ( V ). As we want to differentiate functions, as much as we desire, V will consist of infinitely differentiable functions. As we need to control behavior at infinity to integrate, the elements of V will be compactly supported. So we define V = C ∞ c ( R n ) to be the space of infinitely continuously differentiable functions of compact support. Here recall that the support supp φ of continuous function φ is the closure of the set where φ negationslash = 0; so φ ∈ C ∞ c ( R n ) means that there is a compact (i.e. closed and bounded) subset K of R n such that φ ≡ 0 outside K . As C ∞ c ( R n ) is infinite dimensional, we also need to put a topology on this. Tech nically this means that we should define what open sets are. Rather than doing this (to avoid complexity) we define what convergence of a sequence φ j of functions in C ∞ c ( R n ) means. Definition 1. Suppose { φ j } j =1 , 2 ,... is a sequence in C ∞ c ( R n ), and φ ∈ C ∞ c ( R n ). We say that lim j →∞ φ j = φ if (i) there is a compact set K such that φ j ≡ 0 outside K for all j , (ii) and all derivatives of φ j converge uniformly to φ , i.e. for all multiindices α , sup R n  D α ( φ j − φ )  → 0 as j → ∞ . Explicitly, for all α ∈ N n and ǫ > exists N such that for j ≥ N , sup R n  D α ( φ j − φ )  < ǫ . Lemma 0.1. For all x ∈ R n and ǫ > there is a function φ ∈ C ∞ c ( R n ) such that φ ( x ) > , φ ≥ and supp φ ⊂ { x :  x − x  < ǫ } . Proof. First one checks that the function χ defined by χ ( t ) = e − 1 /t , t > 0; χ ( t ) = 0 , t ≤ , is in C ∞ ( R ). Then we let φ ( x ) = χ parenleftbigg ǫ 2 2 −  x − x  2 parenrightbigg ....
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 Fall '10
 Vasy
 Derivative, Hilbert space, Topological vector space, Linear functional

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