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Thread: Infinite dimensional vector Space

  1. #1
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    Infinite dimensional vector Space

    Show that set of real sequence whose infinite sum is convergent is not a finite dimensional vector space over R
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    Quote Originally Posted by math.dj View Post
    Show that set of real sequence whose infinite sum is convergent is not a finite dimensional vector space over R
    For n=1,2,3,..., let S_n be the sequence with a 1 in the n'th position and zeros everywhere else: S_n = (0,0,\ldots,0,\mathop{1}\limits_{\mathstrut\kern-0.5em {(n\text{'th} \atop \text{coord.})}\kern-0.5em},0,0,\ldots).

    Check that these sequences form an infinite linearly independent set (in the vector space of real sequences with convergent sum).
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    to show that these sequences form a infinite linearly independent set what do i do?
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    Quote Originally Posted by math.dj View Post
    to show that these sequences form a infinite linearly independent set what do i do?
    For a given natural number N, show that the set \{S_1,S_2,\ldots,S_N\} is linearly independent. That shows that the dimension of the space is at least N. Since that holds for all N, it follows that the space is infinite-dimensional.
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    Thank you for your help ...U Rock
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