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all norms are not equivalent
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(Example)
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Let $V$ be the vector space of continuous functions
that are differentiable at $0$ . Then we can define norms $$ \Vert f \Vert = \max_{x\in [-1,1]} |f|, $$ and $$ \Vert f \Vert' = \Vert f \Vert+|f'(0)|. $$ It is not difficult to find a sequence of functions $f_1, f_2, \ldots$ in $V$ such that
- $f_k'(0)=k$ for $k=1,2,\ldots$ ,
- $\Vert f_k\Vert = 1$ .
Then $\Vert f_k \Vert = 1$ , and $\Vert f_k \Vert'=1+k$ , so there is no $C>1$ such that $$ \Vert f \Vert' \le C \Vert f \Vert \quad f\in V, $$ and $\Vert\cdot \Vert$ and $\Vert\cdot \Vert'$ cannot be equivalent.
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Cross-references: functions, sequence, norms, differentiable, continuous functions, vector space
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This is version 3 of all norms are not equivalent, born on 2005-12-06, modified 2006-07-29.
Object id is 7519, canonical name is AllNormsAreNotEquivalent.
Accessed 2189 times total.
Classification:
| AMS MSC: | 46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous) |
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Pending Errata and Addenda
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