hi i have a question:the operator $-\Delta$ with domain $H^{2}(R))$ the question is: if $f\in C_{0}^{\infty}(R)$ prove that f(T) is well defined?
clearly the answer is by using functional calculus but as i know if f \ in C(spectrum($-\Delta$)) the answer is clear but in this case its not clear for me , i think also that may be i should use $L^2$representation (so that the operator is unitarly equivalent to multiplication operator ) so if any one has suggestions ,im very grateful thanks