Square Integrable Vector Space

#Math
The vector space of functions that are Square Integrable

Topics

• Hilbert Space

$\displaystyle L^{2}\left{ f:\mathbb{R}^{d} \right}$

• WIP

$\displaystyle \mathfrak{B}=\left{\sin\left( \frac{n\pi}{L}x \right),n \in\mathbb{N}\right}$

• A basis of this vector space is given above
• Proof of orthogonality:
  $$
  \begin{align}
  \int \sin\alpha\sin \beta , \mathrm{d}x & =\int \frac{1}{2}(\cos \alpha \cos \beta-\cos \alpha \cos \beta+2\sin \alpha \sin beta) , \mathrm{d}x \\ &=\frac{1}{2}\int (\cos \alpha \cos(-\beta)-\sin \alpha \sin(-\beta)-[\cos \alpha \cos \beta-\sin \alpha \sin \beta]) , \mathrm{d}x \\ &=\frac{1}{2}\int \left(\cos(\alpha+\beta)-\cos(\alpha-\beta)\right) , \mathrm{d}x \\ \end{align}
  $$
  ...