Does a continuous embedding preserve gaps between subspaces?

Question

I have a separable, reflexive Banach space $(V,\|\cdot\|)$ that is continuously and densely embedded in a Hilbert space $(H,|\cdot|)$. This means, there is a bounded linear injection map $j\colon V \to H$, i.e. $|j(v)|\leq C_j\|v\|$, and $j(V)$ is dense in $H$. Then, one writes $V\hookrightarrow H$, or simply $V \subset H$, having identified $V$ with $j(V)$.

Now I assume that there are (closed) subspaces $V_1$, $V_2$ of $V$, such that $V = V_1 \oplus V_2$.

QUESTION: Do the assumptions above imply, that the minimum angle between the closures of $j(V_1)=:H_1$ and $j(V_2)=:H_2$ in $H$, that are subspaces of $H$, is bounded away from zero?

I have shown that $H_1 \cap H_2 = \{0\}$, so that - by definition - the cosine of the minimum angle is given as

$$ c:= c(H_1,H_2):=~ \sup ~\{ (h_1,h_2) ~: ~h_1 \in H_1,~ |h_1|=1,~ h_2 \in H_2,~ |h_2|=1 \} .$$

Thus, I want to have, that $c<1$, if $V\hookrightarrow H$, $V=V_1\oplus V_2$, and $H_1 = \overline{V_1}^H$ and $H_2 = \overline{V_2}^H$, where the overline denotes the closure of $j(V_i)$ in $H$, $i=1,2$.

Here are some equivalent conditions for gaps between subspaces.

Let V be a Banach space, and $V_1$ and $V_2$ subspaces with $V_1 \cap V_2 = \{0\}$. Then the statements

1. $ V_1 \oplus V_2$ is closed
2. $V_1^\bot \oplus V_2^\bot \subset V^*$ is closed
3. $\gamma(V_1,V_2):= \inf_{u\in V_1} \frac{dist(u,V_2)}{\|u\|} > 0$

are equivalent. Furthermore, $\gamma > 0$ implies $dist(u,V_2) \geq \gamma \|u\|$, for all $u\in V_1$. The $dist$ function is defined as $dist(u,V)=\inf_{v\in V}\|u-v\|$.

These results can be found in Kato's book Perturbation Theory for Linear Operators in Chapter IV.4. With respect to the minimum angle $c$, one has:

Let $H$ be a Hilbert-space and $H_1 \subset H$ and $H_2 \subset H$ subspaces. Then the statements

1. $c(H_1,H_2) < 1$
2. $H_1 \cap H_2 = \{0\}$ and $H_1 \oplus H_2$ is closed
3. there is a constant $\rho > 0$, such that $|h_1 + h_2| \geq \rho |h_2$, for all $h_1 \in H_1$ and $h_2 \in H_2$
4. $\inf \{ dist(h_1, H_2) : h_1 \in H_1, |h_1|=1 \} > 0$

are equivalent.

These results are given in Galantai's book Projectors and Projection Methods on page 249.

Any help, hint, and counterexample is highly appreciated.

Answer 1

These assumptions are not sufficient.

Suppose that $V_1$ is a Banach space continuously and densely embedded into $H$ by the map $j_1 : V_1 \to H$, with $H_1 := j_1(V_1) \subsetneq H$. (For instance, you could let $H = L^2([0,1])$, and $V_1 = C([0,1])$, or perhaps $V_1 = H^1([0,1])$ if you'd like a Hilbert space). Fix some $y \in H \setminus j_1(V_1)$ (e.g. your favorite discontinuous function in $L^2([0,1])$).

Set $V_2 = \mathbb{R}$ and let $V = V_1 \oplus V_2$ be their direct sum under the norm $\|(x,\alpha)\|_{V}^2 = \|x\|_{V_1}^2 + |\alpha|^2$ (or anything equivalent). Then embed $V$ into $H$ by the map $j(x,\alpha) = j(x) + \alpha y$. $j$ is continuous, and we have $H_1 \subset j(V)$ so that $j(V)$ is dense. But $H_1$ is already dense in $H$, so its closure contains $H_2 := j(V_2) = \mathbb{R} y$. Thus the minimum angle between $H_1$ and $H_2$ is zero.