Linear space vs functional space

Linear spaceFunctional space (on \([a, b]\))
Element\(\mathbf{x} = [x_1, x_2, \dots, x_n]\)\(f = [f(x_i), \dots]\), where \(i \in I\) is an uncountable index on \([a,b]\)
Inner product\(\langle\mathbf{x},\mathbf{y}\rangle = [x_1 y_1, x_2 y_2, \dots, x_n y_n]\)\(\langle f,g\rangle = \int_a^b f(x) \,g(x)\, \mathrm{d}x\)
Orthogonal basis\(\mathbf{x} = \sum_{k=1}^n \alpha_k \mathbf{e}_k\)\(f=\sum_k \alpha_k f_k\)
\(\alpha_k = \langle \mathbf{x},\mathbf{e}_k \rangle\)\(\alpha_k = \langle f, f_k \rangle\)