# A question about equivalence of norms involving infimum

Let $I$ be a Banach space with norm $\lVert\cdot\rVert_I$. The norm $$\inf\{\lVert(G_i(u_i))_i\rVert_{\ell^2}\mid u=\sum_{I \geq 0}u_i\}\qquad\text{is equivalent to}\qquad \lVert{u}\rVert_{I}$$ where the series converges in a separable Hilbert space $X_0$ where $I \subset X_0$ continuously. Here $G_i$ are some given function.

I have seen in papers, that this implies:

$$\lVert(G(v_i))_i\rVert_{\ell^2} \leq C\lVert u\rVert_I$$ where $v_i = (u,\phi_i)_{X_0}\phi_i$ wth $\phi_i$ the o.n. eigenbasis of $X_0$. Why is this true? The inequality is the wrong way for picking a particular function in the infimum, so there must be another way to see this.

For example I saw this in http://arxiv.org/pdf/1404.6195v3.pdf, on section 3.1.3, near the bottom, where they use Theorem 8.2 in that paper.