Square Integrable Vector Space

The vector space of functions that are Square Integrable

Topics

• Hilbert Space

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

• WIP

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

• A basis of this vector space is given above
• Proof of orthogonality:

$$\begin{array}{rl}\int \sin \alpha \sin \beta \mathrm{d}x& =\int \frac{1}{2}(\cos \alpha \cos \beta -\cos \alpha \cos \beta +2\sin \alpha \sin beta)\mathrm{d}x\\ & =\frac{1}{2}\int (\cos \alpha \cos (-\beta )-\sin \alpha \sin (-\beta )-[\cos \alpha \cos \beta -\sin \alpha \sin \beta ])\mathrm{d}x\\ & =\frac{1}{2}\int \left(\cos (\alpha +\beta )-\cos (\alpha -\beta )\right)\mathrm{d}x\end{array}$$

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