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NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here. ...

Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$: $$ \lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\...

This is a complement of my previous question about the sequence spaces (I'm afraid, there will be a third part). Let $\rho:[0,+\infty)^{\mathbb{N}}\to[0,+\infty] $ satisfy the following properties: $...

Given a metric space $(\mathbb{A},d)$ in $\mathbb{R^n}$ with a metric $d$ being the Euclidean metric: If $\lim_{t \rightarrow \infty}||A_{t+1}-A_t||\rightarrow 0$ is a convergent sequence where $A$ ...