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The problem:

Let $X = \{ \phi \in C^{2} [0,L] : \phi (0) = \phi (L) = 0 \}$ and we define an operator $T$ on $X$ by $$ T( \phi (x) ) = - \frac{d^{2}}{dx^{2}} \phi (x) = - \phi '' (x). $$ Then

(a) prove that all the eigenvalues of $T$ are strictly postive;

(b) deduce the orthogonality of the eigenfunctions of $T$;

(c) prove that the set of eigenfunctions of $T$ is a complete orthogonal system of $$ \{f : [0,L] \to \mathbb{R} : f \mbox{ is piecewise continuous } \}.$$

My Question:

So is this ODE problem? It seems related Linear algebra too. I have Friedberg's linear algebra text which contains a small section of DE but I don't think it contains enough for this...

Could you suggest me a good reference?

Note:

This was an entrance examination question for Master degree in math. The university uses the textbook by Braun 'Differential Equations and Their Applications' for their DE course.. Does this book have enough to theory for this kind of math?

And I feel very thankful if you give me hints or solutions.

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2 Answers 2

up vote 1 down vote accepted

This is really the basic motivating example of Sturm--Liouville theory, for which the relevant Wikipedia page gives multiple references. If it looks like linear algebra, it's because Sturm--Liouville theory finds its natural home in the theory of unbounded operators on Hilbert spaces, the relevant infinite-dimensional generalisation of the theory of linear transformations on finite-dimensional inner product spaces.

P.S. Friedberg/Insel/Spence deals with linear ODE with constant coefficients precisely because that theory very quickly reduces to honest-to-goodness finite-dimensional linear algebra on the relevant finite-dimensional subspaces of $C^\infty(\mathbb{R})$.

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The best reference of which I know is Applied Analysis by the Hilbert Space Method by Samuel Holland. He was my professor for this material, and we had proofs of this book as our text. I cannot recommend highly enough.

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