vdgreat
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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 \epsilon V |f(x) = f(−x), \forallx \epsilon R} , and let O be the subset of odd
functions, so that O = {f \epsilon V |f(x) = −f(−x), \forallx \epsilon R} . Prove that:
(a) E and O are subspaces of V .
(b) E \cap O = {0}.
(c) E + O = V .
operations of function addition and scalar multiplication. Let E be the subset of even
functions, so E = {f \epsilon V |f(x) = f(−x), \forallx \epsilon R} , and let O be the subset of odd
functions, so that O = {f \epsilon V |f(x) = −f(−x), \forallx \epsilon R} . Prove that:
(a) E and O are subspaces of V .
(b) E \cap O = {0}.
(c) E + O = V .