# Is this an ODE problem, related to linear algebra? May I get a reference text?

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.

-

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 $C^\infty(\mathbb{R})$.