Hilbert Spaces

April 10, 2026 · Luciano Muratore

Building upon our understanding of finite-dimensional Euclidean space RN\mathbb{R}^N, we now extend these geometric concepts to infinite dimensions. This transition is not merely a mathematical curiosity; it is a necessity for Digital Signal Processing (DSP), where signals often behave like infinite-dimensional vectors.

Motivation: Why This Matters

In DSP, we require a robust mathematical framework where several key properties hold true:

  • Well-defined Energy and Correlation: We must be able to measure the strength and similarity of signals.
  • Convergence: Approximations of signals must converge to a stable limit.
  • Stability: Linear operations performed on signals must remain stable under transformation.

Hilbert spaces provide exactly this structure, effectively generalizing Euclidean geometry to infinite-dimensional contexts.

What is a Hilbert Space?

A Hilbert space is a specialized vector space (typically over R\mathbb{R} or C\mathbb{C}) defined by four critical properties:

  1. Inner Product: It is equipped with an inner product ⟨x,y⟩\langle x, y \rangle.
  2. Induced Norm: It possesses a norm defined by the inner product: ∥x∥=⟨x,x⟩\|x\| = \sqrt{\langle x, x \rangle}.
  3. Completeness: It is a “complete” space, meaning every Cauchy sequence of vectors converges to a limit that is also within the space.

The Space l2(Z)l_{2}(\mathbb{Z})

The space l2(Z)l_{2}(\mathbb{Z}) is the natural home for discrete-time signals with finite energy[cite: 72]. It is defined as the set of all sequences x[n]x[n] that are square-summable:

l2(Z)={x[n]:∑n=−∞∞∣x[n]∣2<∞}l_{2}(\mathbb{Z}) = \left\{ x[n] : \sum_{n=-\infty}^{\infty} |x[n]|^{2} < \infty \right\}

This space is a Hilbert space because it is complete and features a well-defined inner product:

⟨x,y⟩=∑n=−∞∞x[n]y[n]‾\langle x, y \rangle = \sum_{n=-\infty}^{\infty} x[n] \overline{y[n]}

Basis and Representations

Just as RN\mathbb{R}^N has a canonical basis, l2(Z)l_{2}(\mathbb{Z}) has an orthonormal basis consisting of unit impulses {ek}\{e_{k}\} defined by ek[n]=δ[n−k]e_{k}[n] = \delta[n-k]. Consequently, any signal x[n]∈l2(Z)x[n] \in l_{2}(\mathbb{Z}) can be uniquely represented as a linear combination of these basis vectors:

x[n]=∑k=−∞∞x[k]ek[n]x[n] = \sum_{k=-\infty}^{\infty} x[k] e_{k}[n]

The Problem of Completeness: l0l_{0} vs. l2l_{2}

In practical computation, we often work with finite support sequences (l0l_{0}), where x[n]=0x[n] = 0 for all but a finite number of nn. While l0l_{0} is convenient and dense within l2(Z)l_{2}(\mathbb{Z}), it is not complete.

We can prove this by constructing a sequence of finite-support signals yky_k that converges to a signal with infinite support. For instance, if we define yk[n]y_k[n] as 1/n1/n for a finite range kk and 00 elsewhere, as k→∞k \to \infty, the limit y[n]=1/ny[n] = 1/n is an infinite-support signal. Because the limit of this sequence “escapes” the set of finite-support signals, l0l_{0} fails the completeness test required to be a Hilbert space.