let V be ComplexLinearSpace; :: thesis: for V1 being Subset of V
for W being Subspace of V st the carrier of W = V1 holds
V1 is linearly-closed
let V1 be Subset of V; :: thesis: for W being Subspace of V st the carrier of W = V1 holds
V1 is linearly-closed
let W be Subspace of V; :: thesis: ( the carrier of W = V1 implies V1 is linearly-closed )
set VW = the carrier of W;
reconsider WW = W as ComplexLinearSpace ;
assume A1: the carrier of W = V1 ; :: thesis: V1 is linearly-closed
thus for v, u being VECTOR of V st v in V1 & u in V1 holds
v + u in V1 :: according to CLVECT_1:def 7 :: thesis: for z being Complex
for v being VECTOR of V st v in V1 holds
z * v in V1
proof
let v, u be VECTOR of V; :: thesis: ( v in V1 & u in V1 implies v + u in V1 )
assume ( v in V1 & u in V1 ) ; :: thesis: v + u in V1
then reconsider vv = v, uu = u as VECTOR of WW by A1;
reconsider vw = vv + uu as Element of the carrier of W ;
vw in V1 by A1;
hence v + u in V1 by Th32; :: thesis: verum
end;
let z be Complex; :: thesis: for v being VECTOR of V st v in V1 holds
z * v in V1
let v be VECTOR of V; :: thesis: ( v in V1 implies z * v in V1 )
assume v in V1 ; :: thesis: z * v in V1
then reconsider vv = v as VECTOR of WW by A1;
reconsider vw = z * vv as Element of the carrier of W ;
vw in V1 by A1;
hence z * v in V1 by Th33; :: thesis: verum