Have you found ?
Fernando Revilla
I am considering the sequences of continuous functions defined by:
How do I show that is a cauchy-sequence in the normed vector space and how do I examine if the mentioned normed vector-space is a banach space (i know that every finite dimensional normed vector space is a banach space).
is the normed vector space of continuous real-valued functions defined in the closed interval .
Thanks.
Have you found ?
Fernando Revilla
Every finite dimensional vector space is a Banach space just by the "accident" that on finite dimensional vector spaces all norms are equivalent. For infinite dimensional vector spaces such as this isn't always the case. You need to show that the norm induces a complete metric space structure on your vector space. Does it here?
Stupid question: How did you get ?
Since I have shown that:
the norm you are asking for becomes:
I would say that this expression goes to 0 as and tend towards infinity.
Per definition i know that for a sequence to a be a cauchy-sequence it must hold that:
Could I conclude that since the expression above tends towards 0 (which is smaller than ) for going towards infinity, that is a cauchy-sequence?