Consider a separable Hilbert space $\mathcal{H}$. Let $F$ be a vector subspace of $\mathcal{H}$ and let $v\in\mathcal{H}$. If $F$ is of finite dimension $n$, then the distance from $v$ to $F$ can be expressed as $$d(v,F)=\sqrt{\frac{G(v,f_1,...,f_n)}{G(f_1,...,f_n)}},$$ where $(f_1,...,f_n)$ is any basis of $F$ and where for all $k\ge1$ and all $x_1,...,x_k\in\mathcal{H}$, $G(x_1,...,x_k)$ is the determinant of the Gram matrix $(<x_i|x_j>)_{1\le i,j\le k}$.
The above formula doesn't depend on the choice of basis. Is there a convenient "symmetrized" formula that is not expressed as a function of a particular basis but rather depending on all the elements of $F$ in a symmetric fashion?
Is there an analog formula if the dimension of $F$ is infinite?
Thank you in advance.