Real time spectrum analyzer and Fourier Transform
By definition, a real time spectrum analyzer can generate a new spectrum in the time between successive samples. For example, if you sample a signal restricted to 0-1000 Hz at the minimum required sample time of 1/2000 seconds (500 microseconds), and perform an FFT on the collected samples in less than 500 microseconds, that is real time. The spectrum is updated after each new sample. The faster the FFT can be computed, the greater the real time bandwidth. Years ago, FFTs took time measured in millisseconds so the real time bandwidth was very limited.  Today FFTs can be computed in under a microsecond so real time bandwidths measured in MHz are available.