(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have to show that the functional [itex] C_n [/itex] on the space of polynomials on the interval [0,1], that takes the n'th coefficient ie

[tex] C_n\left( \sum_{j=0}^m a_j t^j \right) = a_n [/tex]

is discontinuous with respect to the supremum norm [itex] \|p\|_{\infty} = \sup_{t\in[0,1]}|p(t)| [/itex].

2. Relevant equations

A functional F on a normed space is continuous if and only if

[tex] \sup_{\|x\|\leq 1} |F(x)| < \infty [/tex]

3. The attempt at a solution

Our normed space in the problem is the polynomials with the supremum norm.

As a hint it says to consider

[tex] p_k(t) = \frac{(1-t)^k}{bin(k,n)} [/tex]

bin(k,n) is the binomial coefficient.

The supremum of |p_k(t)| in the interval is attained when t = 0 so

[tex] \|p_k\|_{\infty} = \frac{1}{bin(k,n)} [/tex]

and I get by the binomial formula for (1-t)^k, that

[tex] C_n(p_k) = (-1)^n \|p_k\|_{\infty} bin(k,n)[/tex]

but then I get

[tex] \sup_{\|p_k\|\leq 1} | \|p_k\|_{\infty} bin(k,n) | = bin(k,n) [/tex]

but I think the point with hint was that this was supposed to be infinite so that C_n would be shown to be discontinuous. What am I doing wrong?

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# Homework Help: A discontinuous Functional

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