I am trying to prepare for my exam in analysis, and I am stuck on this old exam question.
"Let $\mathcal{H}$ be a Hilbert space with an orthonormal basis (= maximal orthonormal system) $(e_n)_{n=1}^\infty$. Show that there exists $g,h \in \mathcal{H}$, such that $\langle g,e_n\rangle = \frac{1}{n^{4/3}}$, and $\langle h,e_n\rangle = \frac{1}{n^{2/3}}$. Find $\langle g,h\rangle$."
The hints I've been given is that $\sum_{n=1}^\infty \frac{1}{n^\alpha}<\infty$ for $\alpha>1$, and $\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}$.
I am a little confused by how to do this because it seems to me that the inner product isn't explicitly defined? How can I calculate it, if I don't know anything about what the inner product looks like? A friend of mine said I should use Parseval's identity, but I can't see how it helps me with this problem.
I have found in my textbook that $\langle g,h\rangle = \sum_{n=1}^\infty \langle g,e_n\rangle \overline{\langle h,e_n\rangle}$ for all $g,h\in\mathcal{H}$, and I'm thinking I can somehow use this property.
Hint:
Let $\mathcal H$ be a Hilbert space and $\{e_n\}_{n\geq 1}$ be an orthonormal set. If $(c_n)_{n\geq1}$ is a sequence in $\mathbb C$ such that $\sum_{n\geq 1}|c_n|^2\lt \infty$, then $\sum_{n= 1}^{\infty}c_ne_n$ exists.
To see this, define $$y_k:= \sum_{i=1}^kc_ie_i$$ for all $k\in \mathbb N$. Then $(y_k)_{k\geq 1}$ is a sequence in $\mathcal H$. We have $$\|y_l-y_k\|^2=\left\|\sum_{i=k+1}^lc_ie_i\right\|^2 \leq \sum_{i=k+1}^l\|c_ie_i\|^2=\sum_{i=k+1}^l|c_i|^2$$ for some $l,k\in\mathbb N$ with $l\gt k$. Since $\sum_{n\geq 1}|c_n|^2\lt \infty$, the sum $\sum_{i=k+1}^l|c_i|^2$ can be made arbitrarily small for sufficiently large $l,k\in\mathbb N$. So $(y_k)_{k\geq1}$ is Cauchy in $\mathcal H$ and hence converges.
Now for a start, consider the sequence $\left(\frac1{n^{4/3}}\right)_{n\ge 1}.$ What do you know about $$\sum_{n= 1}^{\infty}{\left(\frac1{n^{4/3}}\right)}^2 ?$$