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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 2

For $1\leq p<q<\infty$, the same finitely supported sequence gives
$$
\frac{\|x^{(N)}\|_p}{\|x^{(N)}\|_q}
=N^{1/p-1/q}\longrightarrow\infty.
$$
One of the two inequalities required for <equivalent norms> therefore fails. Hence
$$
\boxed{\text{no pair }1\leq p<q<\infty\text{ gives Lipschitz-equivalent norms on }V}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 2 / 12E / b / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 2 / 12E / b / ii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 2 / 12E / b / ii / Solution

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