By Harary F., Palmer E.M.

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For this, let φ be a function defined on the line with at least two continuous bounded derivatives with φ(0) = 1 and of total integral equal to one and which vanishes rapidly at infinity. A favorite is the Gauss normal function 2 1 φ(x) := √ e−x /2 2π Equally well, we could take φ to be a function which actually vanishes outside of some neighborhood of the origin. Let φt (x) := 1 x φ . t t 60 CHAPTER 2. HILBERT SPACES AND COMPACT OPERATORS. As t → 0 the function φt becomes more and more concentrated about the origin, but still has total integral one.

We have thus proved convergence in the L2 norm. 2 Relation to the operator d . dx Each of the functions einx is an eigenvector of the operator D= d dx in that D einx = ineinx . So they are also eigenvalues of the operator D2 with eigenvalues −n2 . Also, on the space of twice differentiable periodic functions the operator D2 satisfies (D2 f, g) = 1 2π π π f (x)g(x)dx = f (x)g(x) −π −π − 1 2π π f (x)g (x)dx −π by integration by parts. Since f and g are assumed to be periodic, the end point terms cancel, and integration by parts once more shows that (D2 f, g) = (f, D2 g) = −(f , g ).

The proposition implies that a maximal extension must be defined on the whole space, otherwise we can extend it further. So we must prove the proposition. I was careful in the statement not to specify whether our spaces are over the real or complex numbers. Let us first assume that we are dealing with a real vector space, and then deduce the complex case. 17. THE HAHN-BANACH THEOREM. 33 We want to choose a value α = F (y) so that if we then define F (x + λy) := F (x) + λF (y) = F (x) + λα, ∀x ∈ M, λ ∈ R we do not increase the norm of F .

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Graphical enumeration (AP 1973) by Harary F., Palmer E.M.

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