1. ## Subspaces

Which of the following subsets are subspaces of C[0,1]? Justify
i) { f $\displaystyle \in$ C[0,1] | f(0) = f(1)}

ii){ f $\displaystyle \in$ C[0,1]| f(0) = 0 and f(1) = 0}

iii){ f $\displaystyle \in$ C[0,1]|f(0) = 0 or f(1)=0}

iv){ f $\displaystyle \in$ C[0,1]| f is nondecreasing }

2. Originally Posted by studentmath92
Which of the following subsets are subspaces of C[0,1]? Justify
i) { f $\displaystyle \in$ C[0,1] | f(0) = f(1)}

Yes

ii){ f $\displaystyle \in$ C[0,1]| f(0) = 0 and f(1) = 0}

Yes

iii){ f $\displaystyle \in$ C[0,1]|f(0) = 0 or f(1)=0}

No

iv){ f $\displaystyle \in$ C[0,1]| f is nondecreasing }

No.

Before you write back, sit down and (1) prove the first two are subspaces and (2) then try to show the last two aren't, say by finding functions in C[0,1] which aren't closed wrt sum or multiplication by scalar.

Tonio

3. Originally Posted by tonio
No.

Before you write back, sit down and (1) prove the first two are subspaces and (2) then try to show the last two aren't, say by finding functions in C[0,1] which aren't closed wrt sum or multiplication by scalar.

Tonio
@Tonio - I was trying to do this question. However stuck with terminology. If you can please tell what is meant by C[0,1]? and $\displaystyle f \in$ C[0,1].
Thanks

4. C[0, 1] is the set of functions that are continuous on the interval [0, 1].

i) If f and g are two functions, such that f(0)= f(1) and g(0)= g(1), is that also true for af(x)+ bg(x) where a and b can be any real numbers?

ii) If f and g are two functions, such that f(0)= f(1)= 0 and g(0)= g(1)= 0, is that also true for af(x)+ bg(x) where a and b can be any real numbers?

iii) If f and g are two functions, such that either f(0)= 0 or f(1)= 0 and either g(0)= 0 or g(1)= 0, is that also true for af(x)+ bg(x) where a and b can be any real numbers? (Be careful here. What if f(0)= 0 but $\displaystyle f(1)\ne 0$ and g(1)= 0 but $\displaystyle g(0)\ne 0$?)

iv) If f and g are two nondecreasing functions, is af(x)+ bg(x) also nondecreasing, where a and b can be any real numbers? (Think about a and b being negative.)

5. Originally Posted by aman_cc
@Tonio - I was trying to do this question. However stuck with terminology. If you can please tell what is meant by C[0,1]? and $\displaystyle f \in$ C[0,1].
Thanks

C[0,1] is the set (in fact vector space and in fact algebra) of all real continuous functions defined over [0,1]

Tonio