From Ullman and Hopcroft's Introduction to automata theory, languages, and computation  1ed 1979Consider 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)....The family of languages of space complexity S(n) is denoted by  DSPACE(S(n));Does at most mean taking the maximum or an upper bound?By the definition, assume there is another function S'(n) so thatS'(n)>S(n).If an offline Turing machine M has space complexity S(n),  does M also have space complexity S'(n)?If a language is in DSPACE(S(n)), is it also in DSPACE(S'(n))?Is it incorrect to say that the complexity of an offline Turingmachine is S(n)? Instead should we say that an offline Turingmachine has complexity S(n) or S'(n)?Thanks.
The meaning of at most in the definition of space complexity of a Turing machine
complexity theory