• 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

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: dt04_sub_eng.srt missing

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  • 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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