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Let $K$ be a nondiscrete locally compact topological field, and $V$ be a finite dimensional topological vector space over $K$ and $\{ v_1, \cdots ,v_n\}$ be a base of $V$.

$\varphi : K^n \rightarrow V$ is a linear map $$ \varphi(x_1, \cdots , x_n) = \sum_i x_iv_i . $$ Then, I want to show that $\varphi$ is an open map. (This is a Theorem 3 on page 5 in Basic Number Thory)

Kitamado
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    It may fail for "nondiscrete topological field" such as $\mathbb Q$ with the usual metric. So "locally compact" in your title may be important. – GEdgar May 20 '18 at 12:22
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    First try the case $K = \mathbb R$ with its usual metric. – GEdgar May 20 '18 at 12:50

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