Q1: Let $T: \mathcal{D}(\mathbb{R}^n) \rightarrow \mathbb{R}$ be a linear map. What is in general not correct?

A1: $T(\varphi_1 + \varphi_2) = T(\varphi_1) + T(\varphi_2) $

A2: $T(\lambda \varphi_1 + \varphi_2) = \lambda T(\varphi_1) + T(\varphi_2) $

A3: $T(\varphi_1 + \lambda \varphi_2) = T(\varphi_1) + \lambda T(\varphi_2) $

A4: $T(\varphi_1 + \lambda \varphi_2) = \lambda T(\varphi_1) + \lambda T(\varphi_2) $

Q2: Which two properties define a distribution $T: \mathcal{D}(\mathbb{R}^n) \rightarrow \mathbb{R}$?

A1: It is linear and sequentially continuous.

A2: It is linear and positive.

A3: It is continuous and homogenous.

A4: It is continuous and sequentially continuous.

Q3: What is the notation we use for the set of distributions?

A1: $ \mathcal{D}(\mathbb{R}^n) $

A2: $ \mathcal{D}(\mathbb{R}^n)^\prime $

A3: $ \mathcal{D}(\mathbb{R}^n)^\ast $

A4: $ \mathcal{D}(\mathbb{R}^n)_0 $

Q4: What is the correct definition for the delta distribution?

A1: $ \delta(\varphi) = \varphi(1) $

A2: $ \delta(\varphi) = \varphi^\prime(1) $

A3: $ \delta(\varphi) = \varphi(0) $

A4: $ \delta(\varphi) = \varphi(0) + \varphi(1) $