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Math Help - Inner Product space not induced by L1 norm

  1. #1
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    Inner Product space not induced by L1 norm

    I am trying to find a counterexample to the parallelogram law to show that the L1 norm is not induced by an inner product in [0,1]. I have found one where I can let f(x) = 1 for x in [0,1] and
    g(x) = 1 for x in [1,2]
    However, I want one just where the integral has limits 1 and 0 (i.e. just [0,1]).


    Thanks.
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  2. #2
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    Ok, I have managed to find that f(t) = t and g(t) = 1 - t do not satisfy parallelogram law for the norm of continuous functions in [0,1] to show that this norm is not induced by an inner product.....but after calculating, the parallelogram law is satisfied for these f,g for the L1 norm when I do not want it to be
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