The application of the open mapping theorem

Question

Let $V$ be a Hilbert space and $V_i \subset V(i=1, \ldots, J)$ a number of closed subspaces satisfying $V=\sum_{i=1}^J V_i$, which, by a simple application of the Open Mapping Theorem, implies $$ \sup _{\|v\|=1} \inf _{\sum_i v_i=v} \sum_{i=1}^J\left\|v_i\right\|^2<\infty $$

I don't know how to use the open mapping theorem.

Edit: This claim is made in the paper The method of alternating projections and the method of subspace corrections in Hilbert space by Jinchao Xu and Ludmil Zikatanov. See here.

Answer 1

What I would do is consider the linear map $$ f: V_1 \times \dots \times V_J \to V$$ given by $$ f(v_1, \dots, v_J) = v_1 + \dots + v_J.$$

Notice that:

• The codomain of $f$ is complete since $V$ is a Hilbert space.
• The domain of $f$ is also complete, since $V_1, \dots, V_J$ are closed subspaces of the Hilbert space $V$.
• $f$ is surjective, since $V = V_1 + \dots + V_J$.
• $f$ is clearly continuous.

So it is valid to apply the Open Mapping Theorem to $f$.

The Open Mapping Theorem tells us that there exists a $\delta > 0$ such that for every $v \in V$ with $\| v \| < \delta$, there exists a $(v_1, \dots, v_J) \in V_1 \times\dots \times V_J$ with $\|(v_1, \dots, v_J) \| < 1$ such that $f(v_1, \dots, v_J) = v$.

This is equivalent to saying that there exists a $c \in \mathbb C$ such that for every $v \in V$ with $\| v \| = 1$, there exists a $(v_1, \dots, v_J) \in V_1 \times \dots \times V_J$ with $\|(v_1, \dots, v_J) \| < c$ such that $f(v_1, \dots, v_J) = v$. (For example, take $c = 2\delta^{-1}$.)

But then, for every $v \in V$ with $\| v \| = 1$, $$ \inf_{\sum_{i = 1}^J v_i = v} \sum_{i = 1}^J \|v_i\|^2 < c^2.$$

Hence $$ \sup_{\| v \| = 1} \left( \inf_{\sum_{i = 1}^J v_i = v} \sum_{i = 1}^J \|v_i\|^2 \right) \leq c^2 < \infty.$$