In my opinion it is trivial that if an algorithm is polynomial time in its input length then it is also polynomial space. Is this true and really trivial or do I overlook something? If it is wrong, what counterexamples do we have?
(I know that for decision problems we have $PTIME \subseteq PSPACE$. But I am not sure if this also holds for algorithms of other problems like computing a function, arithmetic operations, etc.)
If a Turing Machine takes polynomial time $p(|w|)$, where $w$ is an arbitrary input string, then the TM uses at most $p(|w|)$ tape cells (one cell for each step). So it takes polynomial space.
Note that a decision problem is a language, or a function of the form $f : \{0,1\}^* \to \{0,1\}$. That is, $f(w) = 1$ if and only if $w$ is in your language.