How can I give two non-equivalent norms to a infinite dimensional vector space with infinite Hamel basis?
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1Consider the vector space of real polynomials, with the $p$-norm, for different values of $p$. – Crostul Apr 18 '16 at 12:54
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Do you want an example of this, or do you want to prove it is true for all infinite-dimensional vector spaces? – GEdgar Apr 18 '16 at 17:51
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I want actually both. – Nina Apr 18 '16 at 17:52
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Let $(X, \|\cdot\|_1)$ be an infinite dimensional normed space. One can show that there exists a discontinuous linear functional, say $L$. Now set $$\|x\|_2 := \|x\|_1 + |L(x)|.$$ Notice that there is no constant $c$ such that $\|x\|_2 \le c\|x\|_1$, since this would imply $|L(x)|\le (c - 1)\|x\|_1$, a contradiction to the assumption that $L$ is not continuous.
Giovanni
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