Solving Vector Spaces Problems with Calculus

[moham_87]

hi

this problem requires calculus, as it also concerns Vector spaces, i solved a lot of Vector spaces problems, either the subset is matrix or ordered pairs.

this question says:
Which of the following subsets of the vector space C(-inf, inf) are subspaces:

(note: C(-infinity, infinity) is vector of functions defined for all real numbers)

- All integrable functions.
- All bounded functions.
- All functions that are integrable on [a,b].
- All functions that are bounded on [a,b].

i just need detailed explanantion if possible...thnx for ur efforts
ur efforts will be appreciated
thank u

 

[Muzza]

Just check the list of things a subspace must fulfil... For example, if f and g are integrable, is f + g? Is tf integrable for all real t? Etc.

 

[moham_87]

thnxxxx a lot Muzza for ur help

wish u good luck
good bye

 

[arildno]

Okay, let's take the second one:
Suppose f and g are bounded.
This means, there exist a number "A" so that |f|<=A for all x.
There exist "B" so that |g|<=B for all x.

But, by the triangle inequality, we have:
|f+g|<=|f|+|g|<=A+B for all x
Hence, f+g is bounded as well.
Can you finish that proof?

Hope this has given you some ideas..

EDIT:
Corrected an equality sign to an inequality sign .