Is the lagrange interpolation polynomials a linear functional?

Question

Im taking numerical analysis and abstract algebra and I think that the Lagrange interpolation polynomials is a linear functional, I did notice that such polynomials are in the dual basis, since it looks like they satisfy the delta-Kronecker that is used in the dual basis.

I am right, or this make no sense.

Thanks.

Answer 1

Dual basis corresponds to a set of linear functionals which take the Kronecker delta on a basis for the vector space. Lagrange interpolation polynomials does an effect of Kronecker delta on the interpolation points, but the interpolation points are not "linearly independent". Therefore it is incorrect to say "the Lagrange interpolation polynomials form a dual basis" in this sense.