By Lars Hormander

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**Example text**

The notion of one-sided limits for real-valued functions arises occasionally, especially when limits are taken as the inputs approach the endpoints of an interval. 2 can be naturally modified in such case, with the only changes occurring regarding which inputs near a are considered. ) We © 2011 by Taylor & Francis Group, LLC 44 Volume 2: Stochastic Equations denote the right-limit at a by lim f (x), meaning that all inputs chosen when forming x→a+ sequences that approach a are comprised of values that are greater than or equal to a.

The notion of an eigenvalue arises in the study of stability theory of ordinary differential equations (ODEs). 15. Let A ∈ MN (R). ) A complex number λ0 is an eigenvalue of A if det(A − λ0I) = 0. ) An eigenvalue λ0 has multiplicity M if det (A − λ0I) = p(λ ) (λ − λ0)M ; that is, (λ − λ0)M divides evenly into det (A − λ0I). a0 , where a, b = 0. ) Compute the eigenvalues of A. ) Compute A−1 and its eigenvalues. ) Generalize the computations in (i) and (ii) to the case of a diagonal N × N matrix B whose diagonal entries are all nonzero.

8. Prove Prop. 9. 9. Let (X , · X ) and (Y , · Y ) be real Banach spaces. 75) 1/2 . 76) Hilbert Spaces Equipping RN with a dot product enhanced its structure by introducing the notion of orthogonality. This prompts us to define the general notion of an inner product on a linear space. 10. Let X be a real linear space. ) x, x X > 0 iff x = 0. The pair (X, ·, ·· X ) is called a (real) inner product space. 49). ) C ([a, b] ; R) equipped with f,g C= ˆ b f (t)g(t)dt. 77). ) W2,m (a, b; R) equipped with f,g W2,k = ˆ b f (t)g(t) + f ′ (t)g′ (t) + .