Hilbert space

Physicaly, an inner product measure the correlation between two signals.

Inner product Let $E$ be a complex vector space. An inner product on $E$ is a sesquilinear form, with hermitian symmetry, positive-definite. i.e. $\langle .,.\rangle : E\times E \rightarrow \mathbb{C}$ is an inner product iff $\forall x,y,z\in E, \langle x + z,y\rangle = \langle x,y\rangle + \langle z,y\rangle$ $\forall x,y,z\in E, \langle x,y+z\rangle = \langle x,y\rangle + \langle x,z\rangle$ $\forall \lambda\in\mathbb{C}\ \forall x,y\in E, \langle \lambda x,y\rangle =\lambda \langle x,y\rangle$ $\forall \lambda\in\mathbb{C}\ \forall x,y\in E, \langle x, \lambda y\rangle = \bar\lambda \langle x,y\rangle$ $\forall x,y\in E, \langle x,y\rangle = \overline{\langle y,x\rangle}$ $\forall x\in E, \langle x,x\rangle = 0 \Rightarrow x = 0$ $\forall x\in E, \langle x,x\rangle \in\mathbb{R}_+$

We can then define the notion of orthogonality between two signals. Orthogonal signals are “blind” to each other and carries some perfectly complementary informations.

Orthogonality Let $E$ be a complex vector space with the inner product u $\langle .,. \rangle$. Two vectors $x,y\in E$ are orthogonal iff $$ \langle x,y \rangle = 0\ . $$

In signal processing, we will mainly use the following inner products:

  • Let $f,g$ two functions of $L^2(\mathbb{R})$. $$ \langle f,g\rangle = \int_{\mathbb{R}} f(t)\overline{g(t)}\mathrm{d} t $$
  • Let $u,v$ two sequences of $\ell^2(\mathbb{Z})$. $$ \langle u,v\rangle = \sum_{n=-\infty}^{+\infty} u_n \overline{v}_n $$

And one can remark that an inner product allows one to define a norm: the 2-Norm: $$ \Vert x\Vert_2 = \sqrt{\langle x,x\rangle} $$

Finally, we can define Hilbert spaces

Hilbert space A Hilbert space is a Banach space with an inner product.

Hence, the $L^2(\mathbb{R})$ and $\ell_2(\mathbb{Z})$ with the inner products given previously are Hilbert spaces.