“讨论室:学术” 上的话题

Determine a map is invertible?

Let $V=V_1\times \cdots \times V_n$ where $V_j$ are Banach spaces. Let $p_i:V\to V_i$ be the projections and $u_i:V_i\to V$ be the inclusion maps. Under what conditions does a continuous linear map $f:V\to V$ have a continuous inverse?

If $V_i=\mathbb{C}$ or $\mathbb{R}$, we know that $f$ is invertible iff $\det (p_{j}f{u}_i)(1)\ne 0$, i.e. $\det (p_{j}f{u}_i)\ne 0$. But for general Banach spaces, what do we require about the maps $p_{j}f{u}_i$? Do we have a similar result?