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}#

1 Answer
Apr 10, 2017

Answer:

#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#