Hilbert Spaces

A Hilbert space is a complete inner product space. We shall denote the inner product associated with Hilbert spaces by $\langle .,.\rangle$. Also, we shall denote the norm generated by this inner product by $\Vert.\Vert$.

For every $p \in \mathbb{R}^+$, we can define the following norm:

Definition: $\Vert f\Vert _p=(\int_{\mathbb{R}}\vert f(x)\vert^p dx)^{1/p}$.

Now we define the following linear space, which we shall use in chapter 3:

Definition: The space $L^p(\mathbb{R})$ is defined as follows:

$L^p(\mathbb{R})=\{f: \Vert f\Vert _p < \infty\}$.

One can show that $L^p(\mathbb{R})$ is a Banach space, for all $1 \leq p < \infty$. A Banach space is a complete normed linear space.