Is the dual space a V* a sub set of V?
They are isomorphic, since they have the same dimension. V*, as the definition says, consists of all linear functionals from V to its scalar field F. It's elements are functions from V to F.v* is a corresponding dual space consisting of all linear functionals on V, the world corresponding is what makes me confused, i can t understand id V and V* are different vector spaces
Note that ##V## and ##V^*## are only isomorphic if ##V## is finite dimensional!!They are isomorphic, since they have the same dimension. V*, as the definition says, consists of all linear functionals from V to its scalar field F. It's elements are functions from V to F.
No, it's not. If E and F are sets, then E is a subset of F if and only if each member of E is a member of F. V can e.g. be a set of of ordered pairs of real numbers, and in that case V* is the set of functions from V into ℝ. Clearly no function from V into ℝ is an ordered pair of real numbers.thank you I believe that I understood, V* and V are isomorphic, so they have same dimension,and a funcional linear is an function of V,so the space V* is a sub set of V,is that right?
There are technicalities of language here. In the first place a vector "space" does have an associated set of vectors, but the "space" is not identical to the set of vectors. (The concept of the "space" includes the set of vectors plus the set of scalars plus the various operations on them.) When you speak of one thing being a subset of another, the two things involved should be sets .so the space V* is a sub set of V,is that right?