I used the recurrence relation to construct Chebyshev polynomials $$ T_0(x)=1 $$ $$ T_1(x)=x $$ $$ T_{n+1}(x)=2x T_n(x) - T_{n-1}(x) $$ for $x\in[-1,1]$
I found that $$ T_0(x) \cdot T_1(x) =0$$ $$ T_0(x) \cdot T_3(x) =0$$ $$ T_0(x) \cdot T_5(x) =0$$ etc. (numerical dot product, programmed in python)
I am pretty sure that $$ T_0 \cdot T_2 =\int_{-1}^1 (2x^2-1)dx \neq 0 $$
in general $$ \int_{-1}^1 T_i(x) T_j(x) dx=0 $$ only holds for $j-i$ is an odd number. Is this correct? I found the same result for Legendre polynomial. So, how they make a basis? This is not what I understand as orthogonal polynomials.