Originally Posted by

**arbolis** I feel sorry to ask help for this problem, since I believe it's a simple one. But I go nowhere when I think alone.

The problem states :

Let $\displaystyle \langle \cdot ,\cdot\rangle$ be one of the following scalar product in the space of polynomials with real coefficients of degree $\displaystyle \leq n$.

a)$\displaystyle \langle f,g \rangle=\sum_{j=0}^n f(x_j)g(x_j)$, with $\displaystyle x_i$ different from $\displaystyle x_j$ when $\displaystyle i$ different from $\displaystyle j$.

b)$\displaystyle \langle f,g \rangle=\int_0^1 f(x)g(x)dx$.

Prove that {$\displaystyle 1,x,x^2,...,x^n$} **is not** an orthogonal set if $\displaystyle n\geq 2$.

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First : I don't know how to start. Second : in the item a), I don't understand why there is the condition of the "i", if it isn't even in the equation. Maybe an error of text, so it should be $\displaystyle f(x_i)g(x_j)$? Third : About what I must prove. It says, an orthogonal set, wouldn't it be an orthogonal basis? But even having said that, I still don't understand at all what they ask me to do. Please help me!