Norms and Cauchy Sequences

Rapha · #1 $Old$ November 24th, 2008, 08:49 PM

Hi!

Two norms $||\bullet ||$ and $|||\bullet |||$ on a vector space V are called equivalent if there exist 0 < a <= b such that

$a ||\bullet || \le |||\bullet ||| \le b ||\bullet ||$

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$||\bullet ||$ and $|||\bullet |||$ are equivalent
Show that

A sequence is a cauchy-sequence on $||\bullet ||$ <=> A sequence is a cauchy-sequence on $|||\bullet |||$

I do not have any clue. Any help would be much appreciated.

best regards
Rapha

Moo · #2 $Old$ November 25th, 2008, 12:40 AM

Hello,

If $(a_n)_{n \in \mathbb{N}}$ is a Cauchy sequence with the $\|.\|$ norm, then write the definition :

$\forall \varepsilon' >0,~ \exists N \in \mathbb{N},~ \text{such that } \forall m,n > N,~ \|x_m-x_n\|< \varepsilon'$

But we know that there exists b such that :
$b \|.\| \geqslant |||.|||$
that is $\|.\| \geqslant \frac 1b \cdot |||.|||$ (which is possible since b>0)

So going back to the definition, we have :
$\frac 1b |||x_m-x_n||| \leqslant \|x_m-x_n\|< \varepsilon'$

Hence :
$|||x_m-x_n|||< b \varepsilon'$

Let $\varepsilon=b \varepsilon'$ and it'll give the Cauchy sequence definition for the $|||.|||$ norm.

Rapha · #3 $Old$ November 25th, 2008, 07:12 AM

Hey Moo,

thank you so much, you are awesome!