Question 1184010: Show that C[a, b], together with the usual scalar multiplication and addition of functions, satisfies the eight axioms of a vector space
Answer by robertb(5830)   (Show Source):
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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,  for any f in C[a,b] and any real constants  and  .
 for any f in C[a,b].
 for any f, g in C[a,b].
Finally,  for all f in C[a,b].
Therefore C[a,b] is a vector space with the usual scalar multiplication and addition of functions.
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