Training Fractionally Spaced Channel Estimator and Equalizer
The training sequence should be at the same spacing as the equalizer when considering its sampling at the input to the equalizer. Adaptive algorithms converge to the least square solution based on the error between the received sequence and the transmitted sequence (known when a training sequence is used). Further, the equalizer can only determine a solution where channel energy is present (for example, if you transmitted a single tone, you would only find the single phase/magnitude weight for that frequency)

If you use only one sample per symbol to derive the channel, the solution will suffer from "band edge aliasing" as I depict in the figures below, and the channel solution will suffer from the folded energy at the band edges. 





For further details see this post where I have derived the operation of the LMS Equalizer and the solution using the Wiener-Hopf equations (for when you have the luxury of block post-processing, but also helps illuminate the operation since this is the same solution that the adaptive algorithms would converge to).

Compensating Loudspeaker frequency response in an audio signal