Exercise 7, § 4

a) With the notations of exerc. 6, show that, for $M + N^{\circ}$ to be closed, it is necessary and sufficient that $M + N^{\circ} = (M^{\circ} \cap N)$. b) Show that the following properties are equivalent : $\alpha) \ \varepsilon(M, N) < 1$; $\beta) \ M + N^{\circ}$ is closed in $E$; $\gamma)$ If $\tilde{M}$ is the orthogonal complement of $M \cap N^{\circ}$ in $M$ and $(N^{\circ})^{\sim}$ that of $M \cap N^{\circ}$ in $N^{\circ}$, then $E$ is the direct sum of $M^{\circ} \cap N, M \cap N^{\circ}, \tilde{M}$ and $(N^{\circ})^{\sim}$. Moreover, if $R$ and $S$ are respectively the projections from $E$ onto $M$ and $(N^{\circ})^{\sim}$ corresponding to this decomposition, then

\| R \| = \| S \| = (1 - \varepsilon^2(\mathbf{M}, \mathbf{N}))^{1/2}. (To see that $\alpha$ implies $\beta$) observe first that if $x \in \tilde{\mathbf{M}}$ and $y \in (\mathbf{N}^\circ)^{\sim}$, then $|\langle x|y \rangle| \leq \varepsilon(\mathbf{M}, \mathbf{N}) \|x\|.\|y\|$; then, let $u = x + y + t$ be the decomposition of an element $u \in \mathbf{M} + \mathbf{N}^\circ$ with $x \in \mathbf{M}$, $y \in (\mathbf{N}^\circ)^{\sim}$ and $t \in \mathbf{M} \cap \mathbf{N}^\circ$, deduce that $\|x\| \leq (1 - \varepsilon^2(\mathbf{M}, \mathbf{N}))^{-1/2} \|u\|, \|y\| \leq (1 - \varepsilon^2(\mathbf{M}, \mathbf{N}))^{-1/2} \|u\|$ and $\|t\| \leq \|u\|$. To prove that $\gamma$ implies $\alpha$), consider the decomposition $v = v_1 + v_2$ for a $v \in \tilde{\mathbf{M}}$, where $v_1$ is the orthogonal projection of $v$ onto $\tilde{\mathbf{N}}$, the orthogonal complement of $\mathbf{N} \cap \mathbf{M}^\circ$ in $\mathbf{N}$; we have $R.v_1 = v$, and if we had $\varepsilon(\mathbf{M}, \mathbf{N}) = 1$, there would exist a sequence $(v_n) \in \mathbf{M}$ such that $\|v_n\| = 1$ and such that $\|(v_n)_1\|$ tends to 0. Next show that the restriction $R_1$ of $R$ to $\tilde{\mathbf{N}}$ is a bijection from $\tilde{\mathbf{N}}$ onto $\tilde{\mathbf{M}}$; to calculate $\|R\|$, show that $\|R_1^{-1}\| \leq (1 - \varepsilon^2(\mathbf{M}, \mathbf{N}))^{1/2}$. c) Deduce from b) that if $\mathbf{M} + \mathbf{N}^\circ$ is closed, then so is $\mathbf{M}^\circ + \mathbf{N}$.