How can Hilbert spaces be used to study the harmonics of vibrating strings? 
The overtones of a vibrating string. 
These are eigenfunctions of an associated Sturm–Liouville problem. 
The eigenvalues 1,1/2,1/3,… form the (musical) harmonic series.
How can Hilbert spaces be used to study the harmonics of vibrating strings?
 A: I'm not qualified to answer this question in detail...but I'd like to point out some things I've learned lately that may be helpful.
Hilbert space is useful when you need an infinite dimensional space to characterize what you are studying and where each mode is orthogonal to the others. Great for Quantum Mechanics...
Where I work, images, that is 2D  matrices of scalar values, that aren't orthonormal at all...can be made into feature spaces.
A feature space can be a cross correlation matrix made from the outer product of all the pixel intensities made into a column, and it's transpose ( a row ).
Then each columns is thought of a orthogonal feature of a manifold in Hilbert space...people play games with these structures but I'm told that one really needs to study more set theory to understand how to process the signal this way.
I do know that complex analysis offers tools for dealing with manifolds but I'm a newbie to all of this.
good luck.
A: Consider an infinite dimensional vector space (what a Hilbert space more or less is).
In this space you can construct any vector as the linear sum of independent unit vectors.
One can consider the vibrating string as just such a vector in the Hilbert space with all overtones as an infinite number unit vectors. The vector is thus a sum of all the overtones.
