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I would just like to know if my following proof is correct:

Claim: If $T:\mathbb{R^n} \to\mathbb{R^m}$ is a linear map, then there exists $C > 0$ such that for every $x \in \mathbb{R^n}\|T(x)\| \le C\|x\|$.

Proof: We have $T(x) = \sum_{i=1}^{n}x_iT(e_i)$, so let $C = n\max(\|T(e_i\|)$. Then,

\|\sum_{i=1}^{n}x_iT(e_i)\| \le \sum_{i=1}^{n}|x_i|\|T(e_i)\|

by the triangle inequality, and

\sum_{i=1}^{n}|x_i|\|T(e_i)\| \le \sum_{k=1}^{n} \max|x_i|\max\|T(e_i)\| \le C\|x\|$$

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    (typo correction:) $\sum\limits_{k=1}^n|x_k|\leq n\|x\|$2012-05-29

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Your proof looks fine. $%And now I avoid the black box bad answer detector just a bit.$

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    +1 for answering the question. Often when the OP provides a proof himself, the question will go unanswered. Ideally, one might try to convince the OP to post his solution as an answer and accept it, but for some reason this rarely happens.2012-05-29