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sequences

bugatti79

Member
Feb 1, 2012
71
Folks,

I am looking at this task.

1) What does it mean to say a sequence converges in a normed linear space?

2) Show that if a sequence fn converges to f in C[0,1] with sup norm then it also converges with the integral norm?

Any idea on how I tackle these?

thanks
 

Plato

Well-known member
MHB Math Helper
Jan 27, 2012
196
1) What does it mean to say a sequence converges in a normed linear space?

2) Show that if a sequence fn converges to f in C[0,1] with sup norm then it also converges with the integral norm?
Do you know about sequence convergence in an ordinary metric space, say the real and/or complex numbers?
If so, then you have the same idea using the norm. After all, is that not how absolute value works?
 

bugatti79

Member
Feb 1, 2012
71
Do you know about sequence convergence in an ordinary metric space, say the real and/or complex numbers?
If so, then you have the same idea using the norm. After all, is that not how absolute value works?
Is it something along the line of

given $\epsilon > 0 $ there exist $ n_0 \in N$ s.t $|(fn-f) (x)|| < \epsilon $ for $n > n_0$ and $ x \in [a,b] $

ie $\forall \epsilon > 0$ there exist $n_0 \in N$ s.t $sup |(f_n-f)(x)|=sup|f_n(x)-f(x)|$ and $x \in [a,b]$ for both.....