When you use the term "frequency space", you are already thinking in terms of the frequency of sin/cos functions. If you actually think in terms of frequency i

When you use the term "frequency space", you are already thinking in terms of the frequency of sin/cos functions. If you actually think in terms of frequency in a different orthonormal basis, you'll find that the dirac delta characteristic you mention is not specific to sin/cos, thereby invalidating your argument.