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all norms are not equivalent

(Example)

Let be the vector space of continuous functions $[-1,1]\to \mathbbmss{R}$ that are differentiable at 0. Then we can define norms

$\displaystyle \Vert f \Vert = \max_{x\in [-1,1]} \vert f\vert,$

and

$\displaystyle \Vert f \Vert' = \Vert f \Vert+\vert f'(0)\vert.$

It is not difficult to find a sequence of functions $f_1, f_2, \ldots$ in

such that

Then $\Vert f_k \Vert = 1$ , and $\Vert f_k \Vert'=1+k$ , so there is no

such that

$\displaystyle \Vert f \Vert' \le C \Vert f \Vert \quad f\in V,$

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 1453 times total.

Classification:

46B99 (Functional analysis :: Normed linear spaces and Banach spaces; Banach lattices :: Miscellaneous)

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