Question 1184010



Let C[a,b] = the set of real-valued continuous functions over the interval [a,b].

Suppose f, g, and h are continuous functions over [a,b].

Since f + (g + h) = (f + g)+ h over [a,b], addition of functions is ASSOCIATIVE.

Since f + g = g + f  over [a,b], addition of functions is COMMUTATIVE.

Since the zero function 0 is continuous over [a,b] and 0 + f = f for any f in C[a,b], 0 is the IDENTITY element for C[a,b].

Since the function -f is also continuous over [a,b] and -f + f = 0, -f is the INVERSE element for any f that is in C[a,b].

Also, {{{alpha(beta*f) = (alpha*beta)f}}} for any f in C[a,b] and any real constants {{{alpha}}} and {{{beta}}}.

{{{1*f = f}}}  for any f in C[a,b].

{{{alpha(f + g) = alpha*f + alpha*g}}} for any f, g in C[a,b].

Finally, {{{(alpha + beta)*f = alpha*f + beta*f}}} for all f in C[a,b].


Therefore C[a,b] is a vector space with the usual scalar multiplication and addition of functions.