Self adjoint compact operators

(bq) part 1 book "functional analysis, sobolev spaces and partial differential equations" has contents: the hahn–banach theorems - introduction to the theory of conjugate convex functions; the uniform boundedness principle and the closed graph theorem; compact operators - spectral decomposition of self adjoint compact operators,...and other contents.

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Example 1.1 Define, for h ∈ R, the operator τh on L2(R) by τhf(x) = f(x − h). Show that τh is bounded. Obviously, τh is linear, and it follows from τhf22 =  +∞ −∞ |f(x − h)|2dx =  +∞ −∞ |f(x)|2dx = f22 , that Tf2 = f2 for all f ∈ L2(R), hence T = 1. Remark 1.1 Here we add that τh is also regular. In fact, if τhf = 0, then f(x−h) = 0 for all x ∈ R, thus f ≡ 0. This shows that τh is injective, hence the inverse operator exists. Then we get by the change of variable y = x − h, i.e....

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