This is the problem:
Let $(H,(\cdot,\cdot))$ be a real Hilbert space and let $\{ e_n \}_{n \in \mathbb{N}}$ a sequence of orthogonal vectors such that for $i \neq j$ $(e_i,e_j)=0$ , where $(\cdot,\cdot)$ denotes the scalar product in $H$. Assume that $\forall x \in H$ we have that the limit
$\lim_{n \to \infty} \sum_{h=1}^{n}(x,e_h)e_h$
exists (i.e is an element of $H$) . Prove that
$sup_{n \in \mathbb{N}} ||e_n||_H$ $ < \infty$
where $|| \cdot||_H$ denotes the norm in $H$.
I am having trouble with this exercise just for the "easy "part I think.My idea was to consider the sequence of operators in dual space $\phi_n(x) \, \colon H \to H $ defined as
$\phi_n(x) = (x,e_n)$.
It is indeed easy to see that,if $H'$ is the dual space, $||\phi_n||_{H'} = ||e_n||_{H}$ by schwartz inequality: So, I thought to apply the uniform boundeness principle to this operator, maybe using orthogonality and the existance of the limit in the hypotesis, but I failed in that as I could not prove that my family of operators satisfied the hypotesis of the theorem. In details I could not prove:
$ \forall x \in H \exists C \in \mathbb{R} \, , C=C(x) $ such that
$sup_{n \in N}||\phi_n(x)||_H \leq C(x)$
I would need a hint to use properly the uniform boundeness principle but even a total solution would be very appreciated.
Consider the sequence of operators in dual space $\phi_n(x) \, \colon H \to H $ defined as $\phi_n(x) = (x,e_n)e_n$. They are clearly continuous linear transformations from $H$ to $H$.
Define $\forall x \in H$, $$T(x) = \lim_{n \to \infty} \sum_{h=1}^{n}(x,e_h)e_h $$
Since $\forall x \in H$ we have that the limit $\lim_{n \to \infty} \sum_{h=1}^{n}(x,e_h)e_h$ exists (i.e is an element of $H$), we have that $\forall x \in H$, $T(x)$ is well-defined and $$ \|T(x)\|_H < + \infty$$
Take $C(x)= \|T(x)\|_H$. Its easy to see that, for all $n$, $$ |\phi_n(x)| = \|(x,e_n)e_n\|_H\leq \|T(x)\|_H =C(x)$$
(Remark: We used the orthogonality of the sequence $\{ e_n \}_{n \in \mathbb{N}}$, to conclude that, for all $n$, $ \|(x,e_n)e_n\|_H\leq \|T(x)\|_H $)
So, by the Uniform Boundness Principle, you have:
$$sup_{n \in \mathbb{N}} \|\phi_n\| < \infty$$
that is
$$sup_{n \in \mathbb{N}} \|e_n\|_{H}^2< \infty$$
that is
$$sup_{n \in \mathbb{N}} \|e_n\|_{H} < \infty$$