Rank of sum of projections

Question

Let $(\varphi_j)$ be a linear independent sequence of elements of a Hilbert space, not necessarily orthogonal, but such that $$Kf := \sum_{j=1}^\infty \langle\varphi_j, f\rangle\varphi_j$$ converges for every $f$ in the Hilbert space. Is it then true that the rank of $K$, i.e., the dimension of its range, is infinite?

Answer 1

I think this is true even if the underlaying space $V$ is not complete, i.e., an inner product space: The coimage $(\ker K)^\perp$ is isomorphic to $\operatorname{ran} K$, since the restriction of $K$ to the coimage is injective. The claim thus follows from $\dim(\ker K)^\perp = \infty$. To show that, let $f\in\ker K$. Then $\sum_{j=1}^\infty\langle\varphi_j,f\rangle\varphi_j = 0$, and thus $$ \Big\langle f, \sum_{j=1}^\infty\langle\varphi_j,f\rangle\varphi_j\Big\rangle = \sum_{j=1}^\infty\lvert\langle\varphi_j,f\rangle\rvert^2 = 0, $$ which implies $\operatorname{lin}\{\varphi_j:j\in\mathbb{N}\}\subseteq(\ker K)^\perp$.