Eigenfunctions of Laplacian and orthonormal basis (with different inner products) Suppose I have $L^2(\Omega)$ which has two inner products that are both norm-equivalent.
The eigenfunctions of the Laplacian $\Delta$ we know forms an orthonormal basis of $L^2(\Omega)$ -- with respect to which inner product? Both of them? How is that? Is it because they're both norm equivalent?
 A: In general, when one talks about orthogonality of eigenfunctions of $\Delta$, the inner product
$$(u,v)_2=\int uv,\ \forall\ u,v\in L^2 $$ is being assumed. 
We can have another inner product with equivalently norm, but in this new inner product the eigenfunctions are not orthogonal, indeed, let $p\in L^\infty$ satisfying $$0<a\leq p$$
Define $$(u,v)_p=\int puv,\ \forall\ u,v\in L^2$$
You can check that $(\cdot,\cdot)_p$ is a inner product in $L^2$ and $|u|=(u,u)_p^{1/2}$ is equivalently to $(u,u)_2^{1/2}$.
A: Here are some additional examples illustrate Tomás' answer that orthogonality doesn't translate among weighted $L^2$-inner product.

A $W$-weighted $L^2$-inner product is:
$$
(u,v)_W := \int_{\Omega} W uv.
$$
Notice that the equivalence of the norm is:
$$
c \|u\|_{L^2(\Omega)} \leq \|u\|_{W} \leq C\|u\|_{L^2(\Omega)},
$$
by definition this is just for $u$:
$$
c\int_{\Omega}  u^2 \leq \int_{\Omega} W u^2 \leq C\int_{\Omega}  u^2,
$$
not for different $u$ and $v$:
$$
c\int_{\Omega} uv \stackrel{?}{\leq } \int_{\Omega} W uv\stackrel{?}{\leq } C\int_{\Omega} uv,
$$
Hence the norm equivalence tells you nothing about orthogonality. 

Example: We can construct the polynomial to our interest by Gram-Schmidt procedure to the polynomial basis set of $L^2(\Omega)$, under $W$-weighted $L^2$-inner product. Then a normalization will give you the orthonormal basis set for $L^2(\Omega)$.
Different weight produces different set of polynomials, and they are only orthogonal under their own weighted inner product that they have been constructed from. 
If we take $\Omega = (-1,1)$, famous examples are:


*

*$W = 1 \longrightarrow$ Legendre polynomials.

*$W = \sqrt{1-x^2} \longrightarrow$ Chebyshev polynomials.

*$W = (1+x)^{\alpha}(1-x)^{\beta}\longrightarrow$ Jacobi polynomials.


Moreover, if the weight $W$ is bounded above and below, then the weighted $L^2$- and unweighted $L^2$-norm are equivalent.

Last remark: Also OP's question reminds me of the eigenfunctions of the Hamiltonian in Schrödinger's equation, for different potential terms, you can have Bessel functions, spherical harmonics, Airy functions, Laguerre functions, ... and the list gones on and on. Different eigenfunctions are orthogonal under different weight.
