Question

Let V=[a,b] be a vector space and let S be a subset of V defined a S={ f € V | integral from a to b of f(x) =0} determine if S is a subspace of V if so prove it

I know that it is a subspace because is satisfies axioms 1 and 6 because it's defined every where on [a,b] but I'm not sure how to write our the proof?

Answer 1

To show that \(S\) is a subspace of \(V\) you only need to satisfy three things:

(1) The zero vector of \(V\) is also in \(S\).

(2) For any \(f,g\in S\), you also have \(f+g\in S\).

(3) For any scalar \(k\) and \(f\in S\), you have \(kf\in S\).

Answer 2

In \(V=[a,b]\), the zero vector is the function \(f(x)=0\), and you have for any real \(a,b\),
\[\int_a^b 0\,dx=0\]
so (1) is satisfied.

Answer 3

Take any two functions in \(S\), and you have
\[\int_a^b(f+g)\,dx=\int_a^bf\,dx+\int_a^bg\,dx=0+0=0\]
so (2) is also satisfied.

Answer 4

And finally, take any scalar \(k\) and function \(f\in S\), and you have
\[\int_a^bkf\,dx=k\int_a^bf\,dx=0k=0\]
so \(S\) is indeed a subspace of \(V\).