I read a book about orthogonal polynomials written by Gabor Szego. There is a conclusion:
Let the real-valued functions $f_0(x),f_1(x),\dots,f_l(x)$, where $l$ is finite or infinite, be of the class $L_\alpha^2[a,b]$ and linearly independent. By orthogonalization, there exists one orthogonal set $\phi_0(x),\phi_1(x),\dots,\phi_l(x)$ such that for $n = 0, 1, \dots,l$, $$ \phi_n(x) = \lambda_{n0} f_0(x) + \cdots + \lambda_{nn} f_n(x), \quad \lambda_{nn} > 0 \tag{1.1} $$ From (1.1), the following explicit representation holds: $$ \phi_n(x) = (D_{n-1}D_n)^{-\frac{1}{2}} D_n(x) \tag{1.2} $$ for which, for $n \geq 1$, $$ D_n(x) = \begin{vmatrix} (f_0,f_0) & (f_0,f_1) & \cdots & (f_0,f_n) \\ (f_1,f_0) & (f_1,f_1) & \cdots & (f_1,f_n) \\ \vdots & \vdots & \ddots & \vdots \\ (f_{n-1},f_0) & (f_{n-1},f_1) & \cdots & (f_{n-1},f_n) \\ f_0(x) & f_1(x) & \cdots & f_n(x) \end{vmatrix} $$ for $n\geq 0$, $$ D_n = [(f_v,f_\mu)]_{v,\mu=0,1,2,\dots,n}>0 $$ are determinants. We write $D_{-1} = 1$ and $D_0(x) = f_0(x)$.
Then how to prove the relation (1.2)?
Any suggestions would be greatly appreciated.