Dimension of f:R->R is infinity

Dimension of f:R->R is infinity

Hint: suppose the dimension is 2, find a function that cannot be a linear combination of the basis functions. Then use induction.

\begin{align*} \begin{array}{c|cc} x & f_1(x) & f_2(x) \\ x_1 & y_{11} & y_{21} \\ x_2 & y_{12} & y_{22} \\ x_3 & y_{13} & y_{23} \\ \vdots &\vdots &\vdots \end{array} \end{align*}

Let $A = [f_1(x) \; f_2(x)]$ , we want to prove $A\beta = f_3(x)$ does not always have a solution. This is easy, the system is extremely over-determined. $A$ can be reduced to: \begin{align*} \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \vdots & \vdots \end{pmatrix} \end{align*}