Question

Determine whether the following are subspaces of C[-1,1]:

(c) The set of continous nondecreasing functions on [-1,1]

I already know how to prove that something is a subspace, but the problem here is in the term nondecreasing which i don't know how to turn it into a "such that" statement to test it

Answer 1

\[f(x)=x\]

\[(-1)\cdot f(x)=-x\]

Answer 2

Try: A function \(f(x)\) is non-decreasing on an interval \(I\) if \(f(a)\leq f(b)\) for all \(a, b \in I\) such that \(a < b\).

Answer 3

i'm not sure if you can see what I wrote, i'm not good with the equation typing buton

Answer 4

What was cut off at the end? I can only read "\(\alpha f(x)\) might be decre...."

By the way, if you want put \(\LaTeX\) in line, type "\.( the stuff you want the be in math format \.)" Just remove the periods.

Answer 5

"\ ( if i want to test for closure under scalar multiplication, i will do the following
if f in C[-1,1] then alpha f in C[-1,1] also f(x) is non-decreasing means that
alpha f(x) might be decreasing when alpha is negative or so ? \)"

Answer 6

this would mean that f is not in c[-1,1] ?

Answer 7

I believe so. Since it isn't closed under scalar multiplication, not a subspace.

Answer 8

hence my counter example above.

Answer 9

from your example, we can write the statement "is non-decreasing" as "f(x) = x is non decreasing" ?

Answer 10

Yes. Assume \(f(x)\) is non-decreasing, then \((-1)f(x)\) is decreasing.

Answer 11

ok, I see. thanks