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