Encyclopedia > Vector space example 3

  Article Content

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 C[a,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 C[a,b], then
(r*f)(x)=r*f(x).
Then C[a,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 C[a,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 C[a,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 C[a,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 C[a,b], and for f in C[a,b] and x in [a,b],
0(x)+f(x)=0+f(x)=f(x) and hence 0+f=f.

4. For f in C[a,b] consider the function -f,
defined by (-f)(c)=-(f(c)). -f is in C[a,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 C[a,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 C[a,b], then r*f is again a continuous function with values in R and hence an element of C[a,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 C[a,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 C[a,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 C[a,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 C[a,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.

            
         



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Canadian Charter of Rights and Freedoms

... interference of government in the lives of people in a free and democratic society by defining these limits. Regarding similarities with the ECHR there are various ...

 
 
 
This page was created in 24.4 ms