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This is an old homework problem from Folland, and I know it has solutions on this website, but I have some questions about the solution provided to us by our TA because there's things about this problem I don't understand. The question is:

Let $H$ be an infinite-dimensional Hilbert space.

a. Every orthonormal sequence in $H$ converges weakly to $0$

b. The unit sphere $S = \left\lbrace x: \|x\| = 1 \right\rbrace$ is weakly dense in the unit ball $B = \left\lbrace x: \|x\| \leq 1 \right\rbrace$

Part $a$ is easy using Bessel's Inequality. For part $b$, here is the solution we were given and what follows are questions about it I have:

Let $x \in B$. Choose $\left\lbrace u_n \right\rbrace$ as in part $a$ (the orthonormal sequence). For each $u_n$, there is $\lambda_n \in \mathbb{R}$ such that $x+\lambda_nu_n \in S$. We estimate $$|\lambda_n| = \|(x+\lambda_nu_n)-x\| \leq ||x+\lambda_nu_n\|+\|x\| \leq 1 - \|x\| < \infty$$. So $x+\lambda_nu_n \rightarrow x$ weakly since $u_n \rightarrow 0$ weakly by part a.

So here are the questions I have:

1) What does it mean for $S$ to be weakly dense in $B$? I'm not understanding why the above shows weakly density (since I don't grasp the definition, the proof will probably make more sense what I know what we're even trying to show).

2) Why is it that for each $u_n$, there is $\lambda_n \in \mathbb{R}$ such that $x+\lambda_nu_n \in S$?

3) Why does the last inequality $||x+\lambda_nu_n\|+\|x\| \leq 1 - \|x\|$ hold? If $x+\lambda_nu_n \in S$, then it has norm $1$, but is the minus sign just a typo? It probably doesn't matter since you just need $|\lambda_n|$ finite but just wanted to ask anyways

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  • $\begingroup$ 3) is just a typo, should be $\leqslant 1 + \lVert x\rVert$. $\endgroup$ – Daniel Fischer May 12 '16 at 18:59
  • $\begingroup$ Ad 2), consider $f_{x,n}(\lambda) := \lVert x + \lambda u_n\rVert$. We have $f_{x,n}(0) = \lVert x\rVert \leqslant 1$, and $f_{x,n}(3) \geqslant \lVert 3u_n\rVert - \lVert x\rVert \geqslant 2$, so by the intermediate value theorem, there is a $\lambda_n \in [0,3)$ with $f_{x,n}(\lambda_n) = 1$. $\endgroup$ – Daniel Fischer May 12 '16 at 19:02
  • $\begingroup$ Ad 1), do you know the weak topology, or are you only working with weakly convergent sequences? $\endgroup$ – Daniel Fischer May 12 '16 at 19:03
  • $\begingroup$ @DanielFischer Our book defines weak topology as "the topology generated by $X^*$ (the dual space of $X$). I'm not sure what's useful about the weak topology for this problem (tbh I still don't fully understand what the weak topology is) $\endgroup$ – Brenton May 12 '16 at 19:49
  • $\begingroup$ Okay. Understanding the weak topology is important, but takes time. Do you know what "$A$ is dense in $X$" means if $X$ is a topological space? $\endgroup$ – Daniel Fischer May 12 '16 at 20:01

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