Consider the following code:
In[22]:= FourierTransform[Exp[I u t],t, \[Omega]]
Out[22]= Sqrt[2 \[Pi]] DiracDelta[u+\[Omega]]
While,
In[25]:= FourierTransform[Sqrt[2 Pi]DiracDelta[u+\[Omega]],\[Omega],t]
Out[25]= E^(-I t u)
In addition the coefficient are supposed to be 2 Pi
instead of Sqrt[2 Pi]