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    Consider the subset A of C([0,1]) consisting of continuous function f with f(0)=f(1)=0

    In (C([0,1]),||\cdot||_1 ) determine whether the following are accumulation points of the set A

    1) g_1(t)=0
    2)g_2(t)=t

    How would i go about solving this type of question?
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    (Original post by gemma331)
    Consider the subset A of C([0,1]) consisting of continuous function f with f(0)=f(1)=0

    In (C([0,1]),||\cdot||_1 ) determine whether the following are accumulation points of the set A

    1) g_1(t)=0
    2)g_2(t)=t

    How would i go about solving this type of question?
    Do you know the definition of an accumulation point? Which of g_1 and g_2 intuitively seems like it could be an accumulation point?
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    (Original post by Mark13)
    Do you know the definition of an accumulation point? Which of g_1 and g_2 intuitively seems like it could be an accumulation point?

    The one i think is more likely to be an accumulation point is g_1 but im really not sure how i go about solving this... the examples i have previously done.. in previous work i have been given an a_n and find the accumulation points through b_n where b_n = |a_n|^\frac{1}{n}

    Im not really sure where i go with this question, but proving which one is and proving why the other isnt.
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    (Original post by gemma331)
    The one i think is more likely to be an accumulation point is g_1 but im really not sure how i go about solving this... the examples i have previously done.. in previous work i have been given an a_n and find the accumulation points through b_n where b_n = |a_n|^\frac{1}{n}

    Im not really sure where i go with this question, but proving which one is and proving why the other isnt.
    The first thing you need to do is go back to your lecture notes/text book and get familiar with what an accumulation point is - feel free to post any questions you have about the definition up here.
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    (Original post by Mark13)
    The first thing you need to do is go back to your lecture notes/text book and get familiar with what an accumulation point is - feel free to post any questions you have about the definition up here.

    I know I have to find two sequences of functions in A one t hat converges to  g_1(t) and the other to  g_2(t) but I really don't know how I go about doing ththis please help!!
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    (Original post by gemma331)
    I know I have to find two sequences of functions in A one t hat converges to  g_1(t) and the other to  g_2(t) but I really don't know how I go about doing ththis please help!!
    Think about an arbitrary function f in A. Then we know that f is continuous, and f(0)=f(1)=0. Also, ||f - g_2||_{\infty} = \sup_{0 \leq x \leq 1} |f(x) - g_2(x)| \geq |f(1)-g_2(1)|.

    Can you see how to use this to show that g_2 isn't an accumulation point of A?

    g_1 is an accumulation point of A - to show this you need to find a sequence of continuous functions, each of which takes the value 0 at x=0 and x=1, which converge to g_1 in the uniform norm.
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    (Original post by Mark13)
    Think about an arbitrary function f in A. Then we know that f is continuous, and f(0)=f(1)=0. Also, ||f - g_2||_{\infty} = \sup_{0 \leq x \leq 1} |f(x) - g_2(x)| \geq |f(1)-g_2(1)|.

    Can you see how to use this to show that g_2 isn't an accumulation point of A?
    I still don't understand how this shows g_2 isn't an accumulation point :s
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    (Original post by gemma331)
    I still don't understand how this shows g_2 isn't an accumulation point :s
    If g_2 is an accumulation point, then there is a sequence of functions f_n in A such that ||f_n - g_2||_{\infty} \rightarrow 0. If you can show that for any function f in A, we have ||f - g_2|| \geq 1, then no such sequence f_n can exist, so g_2 is not an accumulation point.
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    (Original post by Mark13)
    ...
    Just noting that you seem to have switched from the 1-norm to the infinity-norm.
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    (Original post by ghostwalker)
    Just noting that you seem to have switched from the 1-norm to the infinity-norm.
    Thanks for pointing that out, I need to read things more carefully

    In that case, both functions are accumulation points, and to show that a given function g is an accumulation point, you need to find a sequence of functions f_n such that ||f_n - g||_1 = \int_0^1|f_n(x)-g(x)|dx \rightarrow 0.

    Often a good way to find a sequence of functions converging to the required limit is to think graphically. For example, the sequence of functions f_n defined by:

    f_n agrees with g_2 for all x satisfying 0 \leq x \leq 1-1/n, and then linearly intepolating between the points (1-1/n,1-1/n) and (1,0) after that (it's a lot easier to draw this function than give an expression for it)

    will converge to g_2 in the 1-norm.
 
 
 
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