Dual of the Weil norm proof: Difference between revisions

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We know that the norm on V° is the Linf norm, since the norm on M° is the L1 norm. Our goal is to use this to figure out the norm on V.

Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem (~~[http~~://~~www~~.~~math.unl~~.~~edu~~/~~~s-bbockel1~~/~~928~~/~~node25.html http~~://~~www.~~math.unl.edu/~s-bbockel1/928/node25.html]), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.

Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem (https://web.archive.org/web/20201130153651/https://math.unl.edu/~s-bbockel1/928/node25.html), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.

Since M is the column space of Ξ, then ~M is the left nullspace of Ξ. It happens to be the 1D subspace of V° spanned by <1 1 1 ...; -1|. I will call this augmented val J°, since it's the JIP with this augmented coordinate -1 at the end.

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 We know that the norm on V° is the Linf norm, since the norm on M° is the L1 norm. Our goal is to use this to figure out the norm on V.
-Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem ([http://www.math.unl.edu/~s-bbockel1/928/node25.html http://www.math.unl.edu/~s-bbockel1/928/node25.html]), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.
+Let's say that ~M is the subspace of V° which annhilates M; e.g. it consists of all augmented vals which map M to 0. By a useful corollary of the Hahn-Banach theorem (https://web.archive.org/web/20201130153651/https://math.unl.edu/~s-bbockel1/928/node25.html), the space V is isometrically isomorphic to the quotient space V°/~M with the quotient norm on it. We will use this to compute an expression for the norm on V, aka the dual to the Weil norm.
 Since M is the column space of Ξ, then ~M is the left nullspace of Ξ. It happens to be the 1D subspace of V° spanned by &lt;1 1 1 ...; -1|. I will call this augmented val J°, since it's the JIP with this augmented coordinate -1 at the end.