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

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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
 
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bugatti79 said:
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?
 
Plato said:
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...
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
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