Consider only finite-dimensional spaces; I know that infinite-dimensional spaces are different. Let V be a finite-dimensional vector space, V* its dual, and V** its double-dual. I do not understand how V** is "naturally isomorphic" to V? As nearly as I can tell, V** is no better than V*. The answers on this forum are too confusing. Is there anyone kind enough to give me a very detailed answer? I need all the nuts and bolts. I haven't seen an answer on the internet that I have liked.
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3As written this question was essentially a duplicate of the linked post (and several other posts). If you don't understand the answers given there, I suggest you post a new question, linking to one or more of them and specifying precisely what part of the answer(s) you'd like clarified. – Travis Willse May 30 '21 at 00:40
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Let $K$ be the field over which $V$ is a vector space. The point is that elements of $V$ can be interpreted as elements of $V^{}$: for $v \in V$ and $f \in V^$, the expression $f(v)$ is linear in $v$ for each $f$ and* it is linear in $f$ for each $v$. So from $v$ we get a map $V^* \to K$ where $f \mapsto f(v)$ ("evaluating at $v$) and it is linear: $(f+g)(v) = f(v)+g(v)$ and $(cf)(v) = c(f(v))$ expresses linearity of $f \mapsto f(v)$ as a function of $f$, yes? This is how each $v$ in $V$ can be turned into an element of $(V^)^ = V^{}$ and all of $V^{}$ arises like this just once. – KCd May 30 '21 at 00:47
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I appreciate your reply, KCd. – Jason Broadway May 30 '21 at 00:57
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@Jason You might find my answer here to be helpful – Ben Grossmann May 30 '21 at 02:09
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@JasonBroadway Send $v \in V$ to $v^{} \in V^{}$ defined by $v^{*}(w) = w(v)$ for all $w \in V^$. As you can see, I didn't have to make any choices to do this. Thus, it is a natural isomorphism. You should try to think about why this is only an isomorphism in the finite dimensional case. – Charles Hudgins May 30 '21 at 02:46
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1I told you before to ask specific questions and not just dismiss all the people who make serious efforts here. If you call us all useless, then you should start by learning basic people skills ... not to mention the mathematical skill of asking a specific question other than "I haven't seen an answer on the internet that I have liked." That makes you pretty useless. – Ted Shifrin May 30 '21 at 03:14
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Thank you, KCD, Ben, and Charles! You all are wonderful! I am sorry, Ted; I was tired and achy from two days of researching this problem. (My people skills are fine). The V to V** isomorphism sends each v to a unique J(v), which is ready to "eat" all of the f's that belong to V* according to the command given by Charles. I see what was tripping me up. I didn't know what to do with the f's, but they are built into the woodwork of J(v). I was (wrongly) mingling my f's into the V to V** ismorphism; I foolishly tried to play with f's in the J isomorphism. Thanks, everyone. – Jason Broadway May 30 '21 at 15:49