Basics of tempered distributions

[mcdoll]

This is another tracking issue for me, but if anyone is interested in helping out, then I would be very happy.

The goal is to get tempered distributions and the Fourier transform going and this needs several independent results:

1. An efficient characterization of continuity for Fréchet spaces
   Strictly speaking we don't need Fréchet spaces, but only pseudometrizable locally convex spaces.

• [Merged by Bors] - feat(analysis/locally_convex): locally bounded implies continuous #16550
• [Merged by Bors] - feat(analysis/locally_convex): first countable topologies from countable families of seminorms #16595

2. Basic facts about Schwartz functions

• [Merged by Bors] - feat(analysis/schwartz_space): Multiplication of Schwartz function and functions of temperate growth #18649
• [Merged by Bors] - feat(analysis/schwartz_space): the derivative of Schwartz functions #16756
• Multiplication of Schwartz functions as a bilinear map
• Convolution
• 1-d Gaussian (would be nice to have Define Hermite polynomials #15566 for that)
• Compactly supported smooth functions

3. Definition of tempered distributions

• [Merged by Bors] - feat(topology/algebra/module/strong_operator, analysis/normed_space/operator_norm): strong operator topology #16053
• Definition of tempered distributions
• [Merged by Bors] - feat(analysis/schwartz_space): Delta distribution #16638

4. Fourier transform

• Definition
• [Merged by Bors] - feat(analysis/special_functions): the japanese bracket #16491
• As a continuous map on Schwartz space
• The inverse Fourier transform
• Relationship of derivatives and multiplication operators under FT

[ADedecker]

Compactly supported smooth functions

Just commenting to claim that!