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).
- (r*f)(x)=r*f(x).
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.
Later, which was published by Project Gutenberg before Twenty Years
claiming that it is a sequel to The Three Musketeers, it neglects to
and that etext. This etext also, like some novel editions, uses the
and it covers portions of the etexts The Vicomte de Bragelonne and the
personages behind the characters created by Dumas. Although some of them
frequently, and so they were included.
Anne of Austria: (1601-66) Anne was the daughter of Phillip III of
Regent from 1643.html">1643-61 with Mazarin as her prime minister. Modern
evidence beyond rumor exists of a secret marriage between the two, as
her disease did not appear until 1664. She was supposedly in love with
actually his mistress, though many thought so. She was, though, in her
counterpart, he was a clergyman, a Bernais, and like D'Artagnan, he was a
children, and died around 1674. He was a nephew to M. de Treville,
history can tell, involved with the Jesuits. A German named Nickel
Jean-Paul Oliva headed the order.
Athos: Athos was, in real life, Armand de Sillegue d'Athos d'Auteville.
and died in Paris in 1643. He was probably a nobleman, as Athos was, and
captain of the musketeers from 1634.html">1634-1642. Dumas claimed, in the preface
memoirs of the Comte de la Fere, presumably the same memoirs Athos is
1634 where he served with our four heroes' historical counterparts. He
livres, not one hundred and fifty thousand as Dumas claims, and held the
grandson of Henry IV. and Gabrielle d'Estrees. He was jailed in
(with the aid of Athos and Grimaud according to Twenty Years After).
.
On
wordlookup.net
All is still licensed under the GNU FDL.
It uses material from the wikipedia.
|
|