VECTOR SPACES
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Let F(R) denote the set of all functions f : R -> R. Then F(R) with the usual addition and scalar multiplication of functions is a vector space. Let A be the set of all even functions (f(x)=f(-x)) in F(R), B be the set of all odd functions (f(-x)=-f(x)) in F(R). Show that A,B are vector subspaces of F(R) and F(R) is the direct sum of A+B.

F[R] IS VECTOR SPACE
A IS THE SET OF EVEN FUNCTIONS
LET E1 & E2 BE ANY 2 EVEN FUNCTIONS IN A …
THAT IS …E1[X]=E1[-X]….AND…..E2[X]=E2[-X]………………….1
LET …G=PE1+QE2 , WHERE P & Q ARE ANY SCALARS
WE HAVE G[X] = PE1[X]+QE2[X]…………………USING 1
G[X] = PE1[X]+QE2[X]=PE1[-X]+QE2[-X]=G[-X]
HENCE G IS EVEN & IS AN ELEMENT OF A THE SET OF EVEN FUNCTIONS…
HENCE A IS A SUB SPACE ….PROVED….
B IS THE SET OF ODD FUNCTIONS
LET O1 & O2 BE ANY 2 ODD FUNCTIONS IN B …
THAT IS …O1[X]=-O1[-X]….AND…..O2[X]=-O2[-X]………………….1
LET …H=PO1+QO2 , WHERE P & Q ARE ANY SCALARS
WE HAVE H[X] = PO1[X]+QO2[X]…………………USING 1
H[X] = PO1[X]+QO2[X]=-PO1[-X]-QO2[-X]= -H[-X]
HENCE H IS ODD & IS AN ELEMENT OF B THE SET OF ODD FUNCTIONS…
HENCE B IS A SUB SPACE ….PROVED….
TST ….
F(R) is the direct sum of A+B.
WHERE …	F(R) denote the set of all functions f : R -> R
NOT CORRECT …..PLEASE CHECK BACK …
CONSIDER THE FUNCTION ..
F[X]=X+2 ……….IT IS A FUNCTION FROM R TO R
IT IS NEITHER EVEN NOR ODD SINCE
F[X]=X+2 AND F[-X]=-X+2
F[X] IS NEITHER EQUAL TO F[-X] NOR EQUAL TO -F[-X]
SO F[R] COVERS EVEN FUNCTIONS , ODD FUNCTIONS &
ALSO SOME FUNCTIONS WHCH ARE NEITHER EVEN NOR ODD ..
SO TO SAY F[R] IS DIRECT SUM OF A & B IDS NOT PROPER…
A DIRECT SUM B DOES NOT COVER FUNCTIONS OF TYPE X+2 ETC…
AS PROVED ABOVE …