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Do you know about sequence convergence in an ordinary metric space, say the real and/or complex numbers?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?

If so, then you have the same idea using the norm. After all, is that not how absolute value works?

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Is it something along the line ofDo 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?

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.....