Is the Dual Space a Subset of V?

[MonicaRita]

Is the dual space a V* a sub set of V?

 

[Number Nine]

Surely five seconds on google would answer this question.
What is the definition of dual space? What are its vectors?

 

[MonicaRita]

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

 

[Number Nine]

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

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.

 

[MonicaRita]

thanks number nine

 

[micromass]

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.

Note that $V$ and $V^*$ are only isomorphic if $V$ is finite dimensional!

 

[MonicaRita]

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?

 

[Fredrik]

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?

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.

 

[Stephen Tashi]

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 .

One way to properly phrase a question is "If V is a finite dimensional vector space, Is the set of vectors associated with the space V* a subset of the set of vectors associated with V?"

Another way to properly phrase a question is "If V is a finite dimensional vector space, then is V* a subspace of V?".

In either case, I think the technical answer is "No". It is true that V* is isomorphic to a subspace of V (namely V itself). It is true that there is an isomorphism that maps each vector in the set of vectors associated with V* to a vector in V. However, being isomorphic to something is not the same as being identical to it.

(it wouldn't surprise me to find a particular textbook that says something like "In the material that follows, we shall make no distinction between V* and its isomorphic image in V". You may find books that use a notation that glosses over the distinction between a thing and it's isomorphic image.)

 

[MonicaRita]

Thank you, thank you very much,your answer is clear and complete, and I really neededm your feedback about differences between subspace and subset was crucial that I could understand ,thank you a lot

 

[MonicaRita]

thank you everybody

 

[Fredrik]

A subspace of a vector space V is a subset of V that's also a vector space (with the addition and scalar multiplication operations inherited from V). All you have to do to verify that a set S is a subspace of V is that

• S is a subset of V.
• The zero vector of V is a member of S.
• For all scalars a,b and all u,v in S, au+bv is a member of S.

 

[MonicaRita]

Ok, thak you very much