Question

Determine all real possible s, so that `W={f|int_-1^1f(x)dx=s}` is subspace in real function that continue in `[-1,1]` ?

`W={f|int_-1^1f(x)dx=s}`

Answer 1

`0`

Explanation:

If `f_1 in W_s` then `int_a^b f_1 dx = s` and

If `f_2 in W_s` then `int_a^b f_2 dx = s`

but if `W_s` has structure of subspace then

`alpha f_1+beta f_2 = int_a^b(alpha f_1+beta f_2)dx = alpha int_a^b f_1 dx + beta int_a^b f_2 dx = (alpha+beta)s`

so the only feasible value for `s` is `0`

Now, with `s=0`, if `f_1 in W_0` and `f_2 in W_0` then

`alpha f_1 + beta f_2 in W_0`

NOTE:

With `a=b=1` all odd functions (such that `f(x)=-f(-x)`) are contained in `W_0`