Existence of sequence in $\ell^2$ - Uniform Bundedness Principle Let $X$ be a vector space and $f_0,f_1,...:X\rightarrow \mathbb{K}$ linear functionals on $X$ such that for every $x\in X \ |f_0(x)|^2\le C\sum\limits_{n\in\mathbb{N}}|f_n(x)|^2<\infty$. Show that there exists $\{\lambda_n\}_{n\in\mathbb{N}}\in \ell^2$ such that $\|\lambda_n\|_2^2\le C$ and $f_0(x)=\sum\limits_{n\in\mathbb{N}}\lambda_nf_n(x)$.
Probably I should use Uniform Boundedness Principle, but I don't see how to do it.
 A: For $C > 0$, consider the linear map $F \colon X \to \ell^2(\mathbb{N},\mathbb{K})$ given by $\bigl(F(x)\bigr)_n = f_n(x)$. The inequality
$$C\sum_{n\in \mathbb{N}} \lvert f_n(x)\rvert^2 < \infty$$
shows that $F$ is actually well-defined, and the inequality
$$\lvert f_0(x)\rvert^2 \leqslant C\sum_{n\in \mathbb{N}} \lvert f_n(x)\rvert^2\tag{1}$$
ensures that $\ker F \subset \ker f_0$. Let $Y = \operatorname{im} F$. Since $\ker F\subset \ker f_0$, there is a linear map $g \colon Y \to \mathbb{K}$ with $f_0 = g \circ F$. The inequality $(1)$ shows that for $y\in Y$, with an arbitrary $x \in F^{-1}(y)$, we have
$$\lvert g(y)\rvert^2 = \lvert f_0(x)\rvert^2 \leqslant C\sum_{n\in \mathbb{N}} \lvert f_n(x)\rvert^2 = C\cdot \lVert y\rVert_{2}^2,$$
so $g$ is continuous with norm $\lVert g\rVert \leqslant \sqrt{C}$. Since $\ell^2(\mathbb{N},\mathbb{K})$ is a Hilbert space, we know that we can extend $g$ to a continuous linear form $G\colon \ell^2(\mathbb{N},\mathbb{K}) \to \mathbb{K}$ with $\lVert G\rVert = \lVert g\rVert$ even without invoking the Hahn-Banach theorem.
Now we use the Riesz representation theorem to conclude that there is a $\Lambda \in \ell^2(\mathbb{N},\mathbb{K})$ with
$$G(z) = \sum_{n\in \mathbb{N}} \lambda_n z_n$$
for all $z \in \ell^2(\mathbb{N},\mathbb{K})$ and $\lVert \Lambda\rVert_2 = \lVert G\rVert \leqslant \sqrt{C}$.
