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Vector Space (Example 3)In analysis, many function sets have the structure of a vector space; these are often called linear spaces instead of vector spaces. This third example is one such set of functions.
Example III: Consider the set Ca,b of all continuous functions f defined on the closed interval a,b -> R. Define vector addition: - (f+g)(x)=f(x)+g(x).
Define scalar multiplication: If r is a real number and f in Ca,b, then - (r*f)(x)=r*f(x).
Then Ca,b is a vector space over the field R.
Proof
1. Since R is a field, if r,s, in R, then r+s in R.
Then for f,g in Ca,b and x in a,b, f(x)+g(x) in R. The sum of two continuous functions is continuous, and therefore f+g is an element of Ca,b. 2. Since R is a field, if r,s,t in R, then r+(s+t)=(r+s)+t.
Then for f,g,h, in Ca,b and x in a,b, f(x)+(g(x)+h(x))=((f(x)+g(x))+h(x) and therefore (f+g)+h = f+(g+h). 3. Consider the function 0, where for x in a,b, 0(x)=0, 0 being the neutral element from R.
0 is in Ca,b, and for f in Ca,b and x in a,b,
0(x)+f(x)=0+f(x)=f(x) and hence 0+f=f. 4. For f in Ca,b consider the function -f,
defined by (-f)(c)=-(f(c)). -f is in Ca,b since it is defined from a,b to R and continuous. 5. Since R is a field, for r,s in R, r+s=s+r.
Then for f,g in Ca,b and x in a,b, f(x)+g(x)=g(x)+f(x) and hence f+g=g+f. 6. If r in R and f in Ca,b, then r*f is again a continuous function with values in R and hence an element of Ca,b. 7. Since R is a field, if r,s,t in R, r*(s*t)=(r*s)*t.
Then if r,s in R and f in Ca,b, for x in a,b, (r*s*f(x))=r*(s*f(x)) and hence (r*s)*f = r*(s*f). 8. Since R is a field, 1*r=r for all r in R.
If f is in Ca,b, it follows for x in a,b: (1*f)(x)= 1*f(x)=f(x) and hence 1*f=f. 9. Since R is a field, if r,s,t in R then r*(s+t)=(r*s)+r*t.
Then for r in R, f,g in Ca,b, and x in a,b, r*(f(x)+g(x))= (r*f(x)+r*g(x) and hence r*(f+g)=r*f+r*g. 10. Since R is a field, if r,s,t in R, then (r+s)*t=r*t+s*t.
Then for r,s in R, f in Ca,b and x in a,b, we have (r+s)f(x)=r*f(x)+s*f(x) and hence (r+s)*f=r*f+s*f.
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