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Let $\phi_1, \phi_2,\dots$ be a complete orthonormal system in a Hilbert space. Define vectors by

$$\psi_n=C_n(\sum_{k=1}^n \phi_k-n\phi_{n+1})$$

$(n=1, 2, \dots)$.

(i) Show that $\psi_1, \psi_2, \dots$ form an orthogonal system in this Hilbert space.

(ii) Find constants $C_n$ that make $\psi_1, \psi_2, \dots$ into an orthonormal system.

(iii) Show that if $(\psi_n, x)=0$ for all $n=1, 2, \dots$, then $x=0$.

(iv) Verify:

$$||\phi_k||^2=1=\sum_{n=1}^{\infty} |(\psi_n, \phi_k)|^2$$

I'm self-studying some mathematical physics and really quite lost on this problem. I have read the entire chapter on Hilbert spaces, but am still very lost as to where to begin. I'm looking for someone to help me with this problem. Thanks!

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Do you know the definition of orthogonal system? –  Brad Nov 9 '12 at 0:41
    
Is $(\phi_k-n\phi_{n+1})$ an vector with a 1 in the k-th entry and $-n$ in the (n+1)-th entry, zeros elsewhere? –  Alex Nov 9 '12 at 1:47
    
A vector in an arbitrary Hilbert space doesn't necessarily have "entries", per se. For example, the elements of the Hilbert space $L^2[0,1]$ of square-integrable functions on the unit interval are functions, not sequences. Use the definitions of orthogonal system and orthonormal system, together with the properties of an inner product, to show that $\psi_n\cdot \psi_{n'}=0$ for $n\neq n'$. –  Brad Nov 9 '12 at 1:59

1 Answer 1

up vote 0 down vote accepted

Think about the vectors $\phi_1,...,\phi_n,..$ as an ordinary orthonormal basis in a space. Then, for (ii), by the Pythagorean theorem $||\psi_n||=|C_n|\cdot \sqrt{1^2+\ldots+1^2+n^2}$.

For (iii) use that $\phi_1,..,\phi_n,..$ is complete, that is every $x$ can be written as (a limit) of their linear combination.

For (iv), write down the definition of $\psi_n$ and expand the expression.. For this and for (i), all you have to use is $(\phi_k,\phi_k)=1$ and $(\phi_k,\phi_j)=0$ if $k\ne j$.

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