Currently I'm self studying functional analysis, namely Hilbert spaces. In the text, the author gives the following proposition:
Proposition A: Let $\{\varphi_i(t)\}_1^{\infty}$ be a sequence of continuous functions that forms an orthonormal basis of $L_2[a,b]$. Then the system $$ \left\{\varphi_{i}(t)\varphi_j(s)\right\}_{i,j=1}^{\infty} $$ is an orthonormal basis of $L_2([a,b]^2)$.
After a careful read through, I now understand the proof. However, I thought possibly Proposition A could be generalized to the following:
Conjecture 1$^1$: Let $\{\varphi_i(t)\}_1^{\infty}$ be a sequence of continuous functions that forms an orthonormal basis of $L_2[a,b]$. Then the system $$ \left\{\varphi_{n_1}(t_1)\varphi_{n_2}(t_2)\varphi_{n_3}(t_3)\cdots\varphi_{n_m}(t_m)\right\}_{n_1,n_2,n_3,\dots,n_m=1}^{\infty} $$ is an orthonormal basis of $L_2([a,b]^m)$.
Perhaps the question is ill defined. So, any thoughts here would be appreciated.
$^1$ Is this correct terminology, i.e. using conjecture...?