Isomorphism between ${\ell }^2(\mathbb{Z})$ and $L^2([-\pi ,\pi ])$

The Hilbert space of square-summable sequences ${\ell }^2(\mathbb{Z})$ is indeed isomorphic to the space $L^2([-\pi ,\pi ])$. This is a fundamental result in functional analysis and Fourier analysis.

Explanation:

1. ${\ell }^2(\mathbb{Z})$: This is the space of all complex-valued sequences $(a_n{)}_{n\in \mathbb{Z}}$ such that the sum of the squares of their absolute values is finite:$$\sum _{n\in \mathbb{Z}}|a_n{|}^2<\mathrm{\infty }.$$This space is a Hilbert space with the inner product defined as:$$⟨(a_n),(b_n)⟩=\sum _{n\in \mathbb{Z}}{a}_n\overline{b_n}.$$
2. $L^2([-\pi ,\pi ])$: This is the space of all complex-valued square-integrable functions on the interval $[-\pi ,\pi ]$. A function $f$ belongs to $L^2([-\pi ,\pi ])$ if:

This space is also a Hilbert space with the inner product defined as:

Isomorphism:

The isomorphism between ${\ell }^2(\mathbb{Z})$ and $L^2([-\pi ,\pi ])$ is established via the Fourier series expansion. Specifically:

• Given a function $f\in L^2([-\pi ,\pi ])$, its Fourier coefficients $(c_n{)}_{n\in \mathbb{Z}}$ are given by:

These coefficients satisfy:

so $(c_n)\in {\ell }^2(\mathbb{Z})$.

• Conversely, given a sequence $(c_n)\in {\ell }^2(\mathbb{Z})$, the corresponding function $f\in L^2([-\pi ,\pi ])$ can be reconstructed via the Fourier series:$$f(x)=\sum _{n\in \mathbb{Z}}{c}_n{e}^{inx}.$$This series converges in the $L^2$ norm.

Observe that the map $f\mapsto (c_n)$ is an isometric isomorphism between $L^2([-\pi ,\pi ])$ and ${\ell }^2(\mathbb{Z})$. This means it preserves the inner product and the norm. This is a consequence of the Parseval's identity, which states that: