Transactional memory (TM) allows concurrent processes to organize sequences of operations on shared \emph{data items} into atomic transactions. A transaction may commit, in which case it appears to have executed sequentially or it may \emph{abort}, in which case no data item is updated. The TM programming paradigm emerged as an alternative to conventional fine-grained locking techniques, offering ease of programming and compositionality. Though typically themselves implemented using locks, TMs hide the inherent issues of lock-based synchronization behind a nice transactional programming interface. In this paper, we explore inherent time and space complexity of lock-based TMs, with a focus of the most popular class of \emph{progressive} lock-based TMs. We derive that a progressive TM might enforce a read-only transaction to perform a quadratic (in the number of the data items it reads) number of steps and access a linear number of distinct memory locations, closing the question of inherent cost of \emph{read validation} in TMs. We then show that the total number of \emph{remote memory references} (RMRs) that take place in an execution of a progressive TM in which $n$ concurrent processes perform transactions on a single data item might reach $Ω(n \log n)$, which appears to be the first RMR complexity lower bound for transactional memory.