Is the Dimensionality of Vector Spaces the Same for Different Quantum States?

[arpon]

Consider the particle in a box problem. The number of energy eigenbasis is 'countable' infinity. But the number of position eigenbasis is 'uncountable' infinity. x can take any value from the interval [0,L] Whichever basis I choose, shouldn't the dimensionality of the vector space be the same?

 

[dextercioby]

That is true, iff the „eigenstates” are element of the same topological vector space. But the space of the eigenvectors of X is larger than the space of the eigenvectors of H, or, equivalently, the two spectral equations for X and H do not have solutions in the same space.