Per the lecture, the worst-case lower bound on the number of comparisons made by any comparison-based sort on n elements is

Each pairwise comparison can split the space of possible orderings by at most a factor of 2, and there are n! possible orderings of n distinct elements. So any decision tree resolving them needs at least log_2(n!) ≈ n log n levels. Hence Ω(n log n) is the lower bound. Linear or sub-linear bounds are impossible without using non-comparison information.