1
$\begingroup$

Here $C[0,1]$ is the usual vector space of continuous functions $[0,1]\to\mathbb{R}$ with the usual operations.

What is the subspace generated by $\{f\in C[0,1]:f(t)>0\text{ for some t}\}$? Is there a friendly way to describe this space?

My first thought was the functions $f\in C[0,1]$ with $f(t)>0$ for some $t$ and $f(s)<0$ for some $s$, with the zero function of course. So this set is closed under scalar multiplication, however it's not closed under sum.

Any hints?

  • 2
    Haven't you described the entire space? Any function $f$ which is non-zero has a value $t$ for which $f(t)>0$ or $f(t)<0$. Thus $f$ is contained in the subspace you described. – 2017-01-31

2 Answers 2

2

Let $A=\{f\in C[0,1]:f(t)>0\text{ for some t}\}$ and let $B$ the linear hull of $A$. Now let $f \in C[0,1]$.

Case 1: $f(t)=0$ for all $t \in [0,1].$ Then we have $f \in B$.

Case 2: $f(t)>0$ for some $t \in [0,1]$. Then $f \in A$, hence $f \in B$.

Case 3: $f(t)<0$ for some $t \in [0,1]$. Then $-f \in A$, hence $f \in B$.

Conclusion: $B=C[0,1]$.

0

Suppose $f \in C[0,1]$. Let $p(x) = \max(0,f(x))+1$ and $n(x) = \max(0, -f(x))+1$. Then $f = p-n$ and both $p,n $ are strictly positive. Hence $\operatorname{sp} \{ f | f(t) >0 \ \forall t \} = C[0,1]$.

It follows that $\operatorname{sp} \{ f | f(t) >0 \text{ for some } t \} = C[0,1]$.

  • 0
    Why the downvote? – 2017-01-31