Unformatted text preview: k φ n kk x k so that (as k φ n k is nonzero) n 6 k x k and k x k is an upper bound for F . Recalling (Archimedes) that the only subsets of N that are bounded above are ﬁnite, it follows that F is ﬁnite. Note : As noted in class, the existence of norms on X (indeed, on any real/complex vector space) follows easily from Zorn’s Lemma. Remark : As remarked in class, if F ⊂ N is ﬁnite then X does indeed carry a norm k•k F relative to which φ n is continuous whenever n ∈ F : simply let k • k be any norm on X and for x ∈ X deﬁne k x k F = k x k + X n ∈ F  x n  . 1...
View
Full Document
 Spring '09
 LARSON
 Linear Algebra, Vector Space, Hilbert space, Topological vector space, φN

Click to edit the document details