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    Equivalence of norms and closedness of finite-dimensional subspaces

    P-UCLAB04S-06

    Area
    Prelim
    Seen at
    UCLA
    Years
    2004
    Status
    draft
    Contents

    Let \(\|\cdot\|\) be any norm on \(\mathbb R^n\), and let \(\|\cdot\|_2\) be the Euclidean norm.

    1. Prove that there exists \(d>0\) such that \[ \|x\|\leq d\|x\|_2 \] for every \(x\in\mathbb R^n\), and use this to show that \(N(x)=\|x\|\) is continuous in the usual topology on \(\mathbb R^n\).

    2. Prove that there exists \(c>0\) such that \[ \|x\|\geq c\|x\|_2 \] for every \(x\in\mathbb R^n\).

    Hint: use continuity of \(N\) on the sphere \(\{x:\|x\|_2=1\}\).

    1. Show that if \(L\) is an \(n\)-dimensional subspace of an arbitrary normed vector space \(V\), then \(L\) is closed.

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