We just started learning the inner product and projections, and I was wondering what the geometrical or intuitive interpretation of projections under different inner product spaces is (compared to the usual $\mathbb{R}^3$ and $\mathbb{R}^2$ inner product spaces.
e.g.
What does it look like or mean to have two orthogonal polynomials $f_1 (x) = 1-2x, f_2 (x) = 2-x$ is an inner product for $V = C([0,1])$ for $f,g \in V$ is
$$ \langle f_1 | f_2 \rangle = \int_{0}^{1} (1+x)f_1(x)f_2(x) dx$$
Basically, what does it mean for these two polynomials to be "orthogonal"? in this vector space? (is there a geoemtrical interpretation?)