|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
all norms are not equivalent
|
(Example)
|
|
|
Let $V$ be the vector space of continuous functions
![$ [-1,1]\to \mathbbmss{R}$](http://images.planetmath.org:8080/cache/objects/7519/js/img1.png "$ [-1,1]\to \mathbbmss{R}$") 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.
|
Anyone with an account can edit this entry. Please help improve it!
"all norms are not equivalent" is owned by matte. [ full author list (2) ]
|
|
(view preamble | get metadata)
Cross-references: functions, sequence, norms, differentiable, continuous functions, vector space
There is 1 reference to this entry.
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) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
