- #1
vdgreat
- 11
- 0
Let V be the vector space of all functions from R to R, equipped with the usual
operations of function addition and scalar multiplication. Let E be the subset of even
functions, so E = {f [tex]\epsilon[/tex] V |f(x) = f(−x), [tex]\forall[/tex]x [tex]\epsilon[/tex] R} , and let O be the subset of odd
functions, so that O = {f [tex]\epsilon[/tex] V |f(x) = −f(−x), [tex]\forall[/tex]x [tex]\epsilon[/tex] R} . Prove that:
(a) E and O are subspaces of V .
(b) E [tex]\cap[/tex] O = {0}.
(c) E + O = V .
operations of function addition and scalar multiplication. Let E be the subset of even
functions, so E = {f [tex]\epsilon[/tex] V |f(x) = f(−x), [tex]\forall[/tex]x [tex]\epsilon[/tex] R} , and let O be the subset of odd
functions, so that O = {f [tex]\epsilon[/tex] V |f(x) = −f(−x), [tex]\forall[/tex]x [tex]\epsilon[/tex] R} . Prove that:
(a) E and O are subspaces of V .
(b) E [tex]\cap[/tex] O = {0}.
(c) E + O = V .