Proof of $f\chi_{\Lambda_n} \to f\chi_{\Lambda}$ in admissible spaces for compact sets $\Lambda_n$

Question

I have a question regarding a paper I found dealing with the problem of Phase Retrieval and I would like to know how one statement could be proven.

Here is a link to the paper: https://arxiv.org/abs/1604.03163

At first, here is a short introduction of the setting: We know that $(\Lambda,\mathcal{T})$ is a topology and we can write \begin{align*} \bigcup\limits_{n \in \mathbb{N}} \Lambda_n = \Lambda \end{align*} where $\Lambda_n$ is a compact set with $\Lambda_n \subseteq \Lambda_{n+1}$.

Consider an admissable $\mathbb{K}$- vector space $(B_d,\| \cdot \|_{B_d}) \subseteq \text{Map}(\Lambda,\mathbb{K})$ for $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$ defined by the properties, that the set of functions with compact support $B_c^d$ is a dense subset of $(B_d,\| \cdot \|_{B_d})$ and $(B_d,\| \cdot \|_{B_d})$ is solid, i.e. for every $z \in B^d$ and $w \in \text{Map}(\Lambda,\mathbb{K})$ with $|w| \leq |z|$ pointwise, there holds $w \in B^d$ and \begin{align*} \|w\|=\||w|\| \leq \|z\| . \end{align*} In Theorem 3.13 and Theorem A.2 of the linked paper, we use that for every $f \in B^d$ there holds \begin{align*} f \chi_{\Lambda_n} \overset{B^d}{\to} f\chi_{\Lambda}=f \end{align*} but I don't understand why this has to hold (It is stated, that this holds, because $B^d_c$ is dense in $B^d$).

If the set of compactly supported functions is dense in $(B_d,\| \cdot \|_{B_d})$, we only know that there exists a sequence $(g_n)_{n \in \mathbb{N}} \subseteq B^d_c$ with $g_n \to f$, which is not necessarily $(f \chi_{\Lambda_n})_{n \in \mathbb{N}}$.

Does anyone have an idea how this could be proven? I am somehow completely clueless. Thanks in advance.

Edit: One idea I had was using the sequence $(g_n)_{n \in \mathbb{N}}$ as mentioned above, to maybe use an $\frac{\epsilon}{2}$ - argument, meaning that use the estimate \begin{align*} \|f \chi_{\Lambda_n} - f \| \leq \|f \chi_{\Lambda_n} -g_m \| + \| g_m - f \| \end{align*} for all $n,m \in \mathbb{N}$ and choosing $g_m$ accordingly. My problem with this approach was that the support of $g_m$ and $\Lambda_N$ sufficies no relation that I see could be utilized, so the first term on the right-hand side can't be estimated to be $\frac{\epsilon}{2}$ in my opinion.

Second: I completely forgot the assumption that $(B_d,\| \cdot \|_{B_d})$ is a Banach space, sorry. Maybe this helps.

Answer 1

I think the convergence does not hold as stated. (I have not read the paper. Maybe the authors have made a mistake here, maybe this problem can be fixed in their theorems, or there is additional context which makes the claim true.)

We choose $\Lambda=[0,1]$ and $\Lambda_n=\{0\}\cup [\tfrac1n,1]$. As the space $B_d$ we choose the Banach space of bounded functions on $\Lambda$ with the $\sup$-norm.

If we now consider the constant function $f=1$ then we have $$\| f\chi_{\Lambda_n} - f\| = \|\chi_{(0,\tfrac1n)}\| = 1$$ which is a contradiction to the claim.