I wanted to ask if it is necessary to check both discriminant and convergent validity in addition to reliability in a pilot study, given that the sample size is less than 50.
It is always worthwhile to do as much as you can in a pilot study to assess the measurement properties of the indicators you hope to use in the main research. If you have data (even if only for 50 cases) to examine convergent and/or discriminant validity, you should. It might not be strictly "necessary" (studies get published without always reporting such pilot findings), but it is best-practice. It is much better to know at the pilot stage if one or more of your indicators lack validity than to complete the main research and then discover that your results are obscured by indicator failure.