Q1: Let us consider the set of test functions $\mathcal{D}(\mathbb{R}^n) = C_c^\infty( \mathbb{R}^n)$. What do we mean when we write $$ \varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} \varphi $$

A1: The sequence of functions is uniformly convergent.

A2: There is bounded set such that all supports of $\varphi_k$ lie inside it and the sequence of functions is uniformly convergent.

A3: There is bounded set such that all supports of $\varphi_k$ lie inside it and $$D^\alpha(\varphi_k) \rightarrow D^\alpha(\varphi)$$ uniformly for all multi-indices.

Q2: Does the sequence $(\varphi_k)$ with $\varphi_k = 0$ converge in the space $\mathcal{D}(\mathbb{R}^n)$?

A1: Yes.

A2: No.

Q3: Consider the function $\varphi_k \in \mathcal{D}(\mathbb{R}^n)$ given by $$\varphi_k(x) = \begin{cases} 0 &,~ | x | \geq 1 \ \frac{1}{k} \exp\left( - \frac{1}{1 - |x |^2 } \right) &,~ | x | < 1 \end{cases}$$ Which claim is true?

A1: $$ \varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} 0 $$

A2: $$ \varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} \varphi_1 $$

A3: $$ \varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} 1 $$

A4: $$ \varphi_k \xrightarrow[\mathcal{D}]{k \rightarrow \infty} \varphi_2 $$