Introduction to the problem
Let $f$ be a function in a Banach space $F[X]$ and $V[X]$ a subspace of approximants. We denote $\phi$ as the best approximant of $f$ for $V$, and the deviation of this approximation is defined as the norm of the error:
$$\rho_V(f)=||f-\phi||=\underset{\varphi\in V}{\min} ||f-\varphi||$$
In Least Squares Approximation we use the $L_2$ norm defined as
$$\left(\displaystyle\int_{X}|f(x)|^2d\mu(x)\right)^{\frac{1}{2}}$$
Or when $X$ is a finite set, in $l_2$,
$$\left(\displaystyle\sum_{i=1}^{n} |f(x_i)|^2\right)^{\frac{1}{2}},$$
that are induced by their respective inner products.
The best approximant in Least Squares Approximation is caracterized as follows. The least squares approximant of an element $f\in F[X]$ in the subspace $V[X]$ is the element $\phi$ for which
$$\langle f-\phi,h\rangle = 0$$
holds for each $h\in V$.
Since $V$ is a linear space, the best approximant can be expressed as a linear combination of a basis of $V$,
$$\phi = \displaystyle\sum_{i=1}^n a_i\varphi_i,$$
where $\left\{\varphi_1,\ldots,\varphi_n\right\}$ form a basis of $V$. This, in conjunction with the characterization yields to the following system of equations called normal system:
$$\displaystyle\sum_{i=1}^n a_i \langle\varphi_i,\varphi_k\rangle = \langle f, \varphi_k\rangle,\quad k=1,\ldots, n,$$
The coefficients, $a_1,\ldots,a_n$ completly determine $\phi$. Expressed in a matricial form we use $G_n=\left(\langle\varphi_i, \varphi_j\rangle\right)_{i,j=1,\ldots,k,1,\ldots,n}$, the Gram matrix.
Statement
I want to show that the deviation in Least Squares Approximation can be expressed in the following determinantal form
$$\rho_V(f) = \frac{\det G_{n+1}(\varphi_1,\ldots,\varphi_n,f)}{\det G_n(\varphi_1,\ldots,\varphi_n)} $$