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Let $(e_n)_{n\in \mathbb{N}}$ be unit vectors in a normed space X. I search for a sequence $(y_k)_{k\in \mathbb{N}}$ of convex combinations of the $(e_n)_{n\in\mathbb{N}}$, which converge strongly to $0$. Can someone give me an example? I already showed the theorem:

Let $(x_n)_{n\in\mathbb{N}}$ be a sequence in X, which converges strongly to $x\in X$. There is a sequence of convex combinations of the $(x_n)_{n\in\mathbb{N}}$, which converges strongly to $x$.

But now I have to find a specific example. Can someone help me?

Tobi92sr
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2 Answers2

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Consider $X=\ell^2$, and set $$x_n = \tfrac{1}{n}\big(e_1+\cdots+e_n\big).$$ Then $\|x_n\|^2 = \tfrac{1}{n^2}+\cdots +\tfrac{1}{n^2} = n\cdot\tfrac{1}{n^2} = \tfrac{1}{n}\to 0$ and thus $x_n\to 0$. However, $e_n\rightharpoonup 0$ but not strongly.

max_zorn
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  • Is it works in case if we want convergence to z = (0,0,0,0,...) and belongs to $conv${z_k} (where $k = n, ...,\infty$ and $n = 1, 2, 3, ...$)? For better understanding what i mean: https://math.stackexchange.com/questions/4591629/example-of-sequence-of-convex-combinations?noredirect=1&lq=1 – Ouo Doulo Dec 04 '22 at 23:14
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For a general sequence of unit vector this is false: take the standard unit vector basis of $\ell_1$. To make it work, you need to assume that your sequence goes to 0 weakly. Then your question is about the proof of Mazur's lemma. Disclaimer - in general, these convex combinations cannot be found explicitly as the proof relies on the Hahn-Banach theorem.

Tomasz Kania
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