I am reading an article about reflexive spaces, with a specific example. The article mentions inequalities that I haven't been able to get around to. Here's the setup.
The space $X = (\prod_n \mathbb R^n_\infty)_2$ is the $\ell_2$ product of the finite dimensional spaces with their sup-norm. The norm is given by $$ \Vert \mathbf x \Vert = \left( \sum |x_n|_\infty^2\right)^{1/2}, $$ where $\mathbf x=(x_n)$ with $x_n \in \mathbb R^n$ whenever this is finite. I can show that this space is reflexive and not uniformly convex.
There is a totally separated sequence $ \{ \mathbf x_k \}$ (with $\Vert \mathbf x_k-\mathbf x_j\Vert>\delta>0$) whose elements' norms are monotonically decreasing to 1. Since it is bounded, it converges weakly to $\mathbf x_0$ and we can consider the differences $\mathbf y_k = \mathbf x_k-\mathbf x_0$, which converge weakly to $\mathbf 0$.
Then for every $\varepsilon$ we can find two indices $k,j$ whose terms satisfy the following:
- $\Vert \mathbf x_0\Vert^2+\Vert\mathbf y_k\Vert^2\geq 1+\varepsilon$ and the same for $\mathbf y_j$.
- $\Vert \mathbf y_k + \mathbf y_j \Vert^2 \leq (1+\varepsilon)(\Vert \mathbf y_k\Vert^2+\Vert\mathbf y_j\vert^2)$.
- $\Vert\mathbf x_0 + \frac{\mathbf y_k + \mathbf y_j}{2}\Vert^2 \leq(1+\varepsilon)(\Vert\mathbf x_0\Vert^2+\Vert\frac{\mathbf y_k + \mathbf y_j}{2}\Vert^2)$.
I haven't been able to understand how these inequalities take place. Is it something specific of the norm? or reflexivity? or uniform convexity? Any hints or pointing in the right direction will be greatly appreciated.
Thank you and have a nice day.