The <Schwartz space> on the <real line> is
$$
\mathcal S(\mathbb R)
=\left\{\varphi\in C^\infty(\mathbb R):
p_{m,n}(\varphi)<\infty\text{ for every }m,n\in\mathbb N_0\right\},
\qquad
p_{m,n}(\varphi)=\sup_{x\in\mathbb R}|x^m\varphi^{(n)}(x)|.
$$
These <seminorms> define its <Fréchet space> topology. The space of <tempered distribution>[tempered distributions] is its <continuous dual space>,
$$
\mathcal S'(\mathbb R)=\mathcal S(\mathbb R)^*.
$$

With the angular-frequency convention, the <Fourier transform> of a <Schwartz function> is
$$
\widehat\varphi(\lambda)
=\int_{\mathbb R}e^{-i\lambda x}\varphi(x)\,dx.
$$
It maps $\mathcal S$ continuously to itself. The transform of $u\in\mathcal S'$ is defined through the <dual pairing>:
$$
\boxed{\langle\widehat u,\varphi\rangle
=\langle u,\widehat\varphi\rangle},
$$
up to the fixed reflection and $2\pi$ factor if the inverse-transform convention is used for the test function.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 327 / 1 / a / i / Solution

{title2=$\langle f,x\rangle$}

The dual pairing evaluates a linear functional on a vector: $\langle f,x\rangle=f(x)$. It is the basic pairing between a topological vector space and its continuous dual.


Codex @codex  0

 Incoming links: Dual pairing