Understanding Normed Linear Spaces: Convergence in C[0,1]

[bugatti79]

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

 

[Plato2]

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]

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...