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N. Let x be the n-tuple x = (x1 , . . , xn ). Then x is just a point in the n-dimensional Euclidean space, x ∈ Rnx . Let us introduce the dual space Rnp whose points are n-tuples p = (p1 , . . , pn ). 65) Rn where dx = dx1 . . dxn , and p, x = form is then deﬁned by n j=1 f (x) = (F −1 fˆ)(x) = Rn pj xj . The inverse Fourier trans- dp i e (2π)n p,x fˆ(p) . 66) 14 1 Background in Analysis The n-dimensional delta-function can be represented as the Fourier integral δ(x) = Rn dp i e (2π)n p,x .

1 First Order Partial Diﬀerential Equations Partial diﬀerential equations are usually much more complicated than ordinary diﬀerential equations, even systems of ordinary diﬀerential equations. However, there is one class of partial diﬀerential equations that is as diﬃcult to solve as a system of ordinary diﬀerential equations. These are linear ﬁrst-order partial diﬀerential equations. Let xi , i = 1, . . , n, be the coordinates in the Euclidean space Rn and i ξ (x) be a vector ﬁeld in Rn . Let f be a given smooth function in Rn and L be a ﬁrst-order partial diﬀerential operator of the form n ξ i (x) L= i=1 ∂ .

Similarly, let M be an open set in Rn (in particular, M can be the whole n R ) and μ be a positive function on M . Then the inner product can be deﬁned by dx μ(x) f (x)h(x) . 131) M This deﬁnes the Hilbert space L2 (M, μ) of complex valued functions with ﬁnite L2 -norm. The spaces of smooth functions are not complete and, therefore, are not Hilbert spaces. This is a simple consequence of the fact that there are convergent sequences of smooth functions whose limit is not a smooth function. The L2 spaces of square integrable functions are obtained by adding the limits of all convergent sequences to the spaces of smooth functions.