a function of a real space has uncountable degrees of freedom
Right—that’s exactly the misunderstanding I was addressing in my earlier comment.
An arbitrary function does indeed have uncountable degrees of freedom, but in that context you’re notconsidering it as an element of a Hilbert space. (Those degrees of freedom do not correspond to basis vectors.)
a function of a real space has uncountable degrees of freedom
Right—that’s exactly the misunderstanding I was addressing in my earlier comment.
An arbitrary function does indeed have uncountable degrees of freedom, but in that context you’re notconsidering it as an element of a Hilbert space. (Those degrees of freedom do not correspond to basis vectors.)