# sequences

#### bugatti79

##### Member
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
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
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.....