• Title: Space of Distributions

• Series: Distributions

• YouTube-Title: Distributions 4 | Space of Distributions

• Bright video: https://youtu.be/0QrNkB09hYE

• Dark video: https://youtu.be/_JUKvtPTz00

• Subtitle on GitHub: dt04_sub_eng.srt missing

• Timestamps (n/a)
• Subtitle in English (n/a)
• Quiz Content

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

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