1

The question is stated in the title, but I don't see how to prove it. Each element of $span\{x,Y\}$ is of the form $cx + y$ where c $\in \mathbb{R}$ and $y \in Y$. If I could show that for any convergent sequence in span{x,Y} $\{c_nx + y_n\}$, $c_n$ converges to some $c \in \mathbb{R}$ and $y_n$ converges to some $y \in Y$, I would be done. But I'm prevented from doing that because $c_n$ and $y_n$ are interdepedent. Any ideas?

  • Hint: $X \to X/Y$ is continuous –  Nov 08 '13 at 21:07
  • If you're familiar with quotient spaces, consider $X/Y$ and the space spanned by $\pi(x)$ in that. Otherwise, suppose you had a sequence $c_nx + y_n \to 0$ with $\lvert c_n\rvert + \lVert y_n\rVert = 1$ and derive a contradiction. – Daniel Fischer Nov 08 '13 at 21:08
  • @DanielFischer I'm not familiar with quotient spaces in this context, so I'm more likely to get the contradiction method. But why would we be allowed to make the assumption that there is a sequence $c_nx + y_n \rightarrow 0$ with $|c_n| + ||y_n|| = 1$? –  Nov 08 '13 at 21:15
  • @user106479 We make that assumption to derive a contradiction, which shows that the assumption was false. To show that a Ravenous Bugblatter Beast of Traal doesn't exist, you may assume that one existed an derive a contradiction from that. – Daniel Fischer Nov 08 '13 at 21:20
  • @DanielFischer Right, sorry, I phrased that last comment incorrectly. I don't see how showing that that sequence can't exist will help us prove closedness of the span. –  Nov 08 '13 at 21:31
  • Proving that shows there is a $\delta > 0$ such that $\lVert cx + y\rVert \geqslant \delta (\lvert c\rvert + \lVert y\rVert)$. Do you see how that helps? – Daniel Fischer Nov 08 '13 at 21:33
  • Ah, I believe I've got it. Thanks for your help! –  Nov 08 '13 at 21:56

0 Answers0