Question

V is the set of all the even functions that pass through the origin. V = { f ∈ F (R, R): f (−x) = f (x) for all x and f(0)=0 }

W is the set of all odd functions. W = { g ∈ F (R, R): g(-x) = -g(x) for all x }

Find an isomorphism T : V → W and prove that T is an isomorphism.

Answer 1

8:14 < Q + 8 5 TODO of U= {f: fl-m) = find in and fol- o} ws?9: 91-) --g@mswa } Define Tivow as follows: T(f) = g where glu) = {-fing, nao oT is well-defined i g defined in this way is odd and hence gew. • TV 1-1 Consider T(6) = T(62) → S()= ) + x 20 -fu) = O Haco & f(n) = f(nd #ner Tis ento Toke ge W and define g(x) = { gen), azo gimi, nao Then § is odd and gal=0 * T is onto Also ig) = g . T is a homomorphism Take aer, feu T(&$) = AT® Taf)(x)= xf, a 30 = x S ) 130 = T()) 1-(-), XC Take bo, be su T(2 + bz) (n) = {(f, +1₂) (+) 230 18,+b)) juca el hon) + fein), no l-fil-fatty , #Co = T(,)(-) + T(8) ) :- T itls) = tf) + Illy) berce Tisan ismorphic

Answer 2

va fff FCRIP) fen) = fra) & 10) = o) W= {ge Feie | gew) argany } . he define T: Now such that T (fo)] = fall of ev. Here of is even and offo)20 then fol a ga) is an odd - function and to) ago) E W. F is one one; filter, T41)= T(4) s film zf.ca => fan a ficus su + T is ownone. tuot T is outo: t g(n) E W J f(x)= aga eV such that T(fo]: ft. *99) >8%) A T is outo & T is Isomorphism.