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

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- #1

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

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What is the definition of dual space? What are its vectors?

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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.

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thanks number nine

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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.

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- #8

Fredrik

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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

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Stephen Tashi

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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 aso the space V* is a sub set of V,is that right?

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

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.)

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thank you everybody

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Fredrik

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- 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.

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Ok, thak you very much