From Ullman and Hopcroft's Introduction to automata theory, languages, and computationConsider the off-line Turing machine M of Fig. 12.1. M has a read-only  input tape with endmarkers and k semi-infinite storage tapes. If for  every input word of length n, M scans at most S(n) cells on any  storage tape, then M is said to be an S(n) space-bounded Turing  machine, or of space complexity S(n).   The language recognized by M is also said to be of space complexity S(n).Consider the multitape TM M of Fig. 12.2. The TM has k two-way  infinite tapes, one of which contains the input. All tapes, including  the input tape, may be written upon. If for every input word of length  n, M makes at most T{n) moves before halting, then M is said to be a  T{n) time-bounded Turing machine, or of time complexity T(n). The  language recognized by M is said to be of time complexity T(n).If I am correct, a language can be the one recognized by several Turing machines. Do the last sentences in the two paragraphs assume that for a language, all of its recognizer TMs have the same time complexity and the same space complexity? Is the assumption true? If the assumption is not true, do the sentences imply taking supremum of the complexities over all of the recognizer TMs of the language?Thanks.
Do the time and space complexities of a language depend on the Turing machine which is its recognizer?
complexity theory