I am really confused about this. I know that functions of definite momentum and position can be used as basis and all wave functions can be written in terms of these basis functions. But do the basis function need to be in the space for which it is a basis function? As the function for states of definite momentum are non realistic, they will not be a part of the space of realistic wave functions. But then, can they be used as a basis for that space?
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This answer of mine and the links in it may be of use to you (these states are what I call a "rigged basis" there) – ACuriousMind Oct 07 '19 at 20:03
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@ACuriousMind I am quite new to QM. But, after reading your answer I came to the conclusion that the functions for states of definite momentum are in fact a "rigged" basis for the space of realistic wave functions but they do not lie in that space themselves, but instead lie in a space called the 'Rigged Hilbert Space" which is different from the previous space. Did I get it right? – SaptarshiS Oct 07 '19 at 20:25