• Title: Distributions Form a Vector Space

  • Series: Distributions

  • YouTube-Title: Distributions 7 | Distributions Form a Vector Space

  • Bright video: https://youtu.be/EzUaKFGEvqY

  • Dark video: https://youtu.be/_dTQGnFd8Xk

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: dt07_sub_eng.srt missing

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  • Quiz Content

    Q1: What is correct for the distribution $T = 2 \cdot \delta$?

    A1: $T(\varphi) = \varphi(0)$

    A2: $T(\varphi) = 3 \cdot \varphi(0)$

    A3: $T(\varphi) = 2 \cdot \varphi(0)$

    A4: $T(\varphi) = \frac{1}{2} \cdot \varphi(0)$

    Q2: What is correct for the distribution $T = 2 \cdot \delta + 3 \cdot T_f$ with a locally integrable function $f$?

    A1: $T(\varphi) = 2 \cdot \varphi(0)$

    A2: $T(\varphi) = 2 \cdot \varphi(0) + \int_{\mathbb{R}^n} f(x) \varphi(x) dx $

    A3: $T(\varphi) = 2\varphi(0) + 3\int_{\mathbb{R}^n} f(x) \varphi(x) dx $

    Q3: Consider the duality pairing $\langle \cdot, \cdot \rangle$. What is a correct reformulation? $$\langle T + 2 S, \varphi + 3 \psi \rangle = $$

    A1: $2 \langle T , \varphi + \psi \rangle$

    A2: $2 \langle T-S, \varphi + \psi \rangle$

    A3: $\langle T + 2 S, \varphi + 3 \psi \rangle + \langle S, \varphi + 3 \psi \rangle$

    A4: $\langle T, \varphi + 3 \psi \rangle + 2\langle S, \varphi \rangle + 6\langle S, \psi \rangle$

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