From Ullman and Hopcroft's Introduction to automata theory, languages, and computation
Consider 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.