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Definition 12.3.1   Let $\mathcal{V}$ be an $\mathbb{R} -$vector space with an inner product. Two vectors $\overrightarrow{x} $ and $\overrightarrow{y} $ are orthogonal if their inner product $\langle \overrightarrow{x} ,\overrightarrow{y}\rangle$ is equal to 0.

Example 12.3.2   Let $\mathcal{V}=\mathbb{R} ^2$ with the ``regular'' inner product defined in Example  1.3.

Example 12.3.3   Let $\mathcal{V}$ be the space of all the polynomials in one variable x with coefficients in $\mathbb{R} $, with inner product:

\begin{displaymath}\langle f ,g \rangle = \int_0^1 f(x)g(x) dx

If f1(x)=x and f2(x)=x2, then $\langle f ,g \rangle = \int_0^1 x^3 dx = \left[ \frac 14 x^4 \right]_0^1 = \frac 14$. Thus the functions f1 and f2 are not orthogonal vectors in $\mathcal{V}$.

Example 12.3.4   Let $\mathcal{V}= \text{Span}( \cos x, \sin x )$, with the inner product

\begin{displaymath}\langle f ,g \rangle = \int_{- \pi}^{\pi } f(x)g(x) dx

Then the sine and the cosine function are orthogonal.

Noah Dana-Picard