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## Main Question or Discussion Point

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 .