Question

Find a Hilbert space \(H\) and its subspace \(F\) such that \(H \neq F+F^{\perp}\). This shows that the assumption of closedness in Theorem \(1.33\) is crucial. Hint: Consider the subspace \(F\) of finitely supported vectors in \(\ell_{2} .\) Then \(F^{\perp}=\\{0\\}\) because given \(x \in H \backslash\\{0\\},\left(x, e_{i}\right) \neq 0\) for \(i \in \operatorname{supp}(x) .\)

Short answer

In \ \ell_{2}\, the subspace of finitely supported vectors \ F \ and its orthogonal complement \( F^{\perp} \) such that \(H = F + F^{\perp}\) does not include all vectors in \ \ell_{2}\. This shows the necessity of closedness.

Step-by-step solution

Understanding the Hilbert Space

Consider the Hilbert space \(\ell_{2}\). This is the space of square-summable sequences, meaning each vector \(x \) in \(\ell_{2}\) is represented as a sequence \( (x_{1}, x_{2}, x_{3}, \dots) \) where \(\sum_{i=1}^{\infty}| x_{i} |^{2} < \infty\)\.

Define the Subspace \( F \)

Define the subspace \(F\) of \( \ell_{2} \) as the set of finitely supported vectors. This means that each vector in \(F \) has only a finite number of non-zero components.

Calculate \( F^{\perp} \)

Next, find the orthogonal complement of \(F\). By definition, \(F^{\perp} = \left\{y \in \ell_{2} \mid \ \langle y, x \rangle = 0 \ \forall x \in F\right\}\). It can be shown that \(F^{\perp} =\{0 \} \) because any non-zero vector in \ \ell_{2}\ has a non-zero inner product with some vector in \ F \.

Combine \( F \) and \( F^{\perp} \)

Combine \( F \) and \( F^{\perp} \). According to the hint, \( H = F + F^{\perp} \) would imply that every vector in \( H \) can be written as the sum of a vector in \( F \) and a vector in \( F^{\perp}\). However, since \( F^{\perp} = \{0\} \), this implies \( H \) must be equal to \( F \).

Disprove \( H = F + F^{\perp} \)

To show \( H \eq F + F^{\perp}\), note that \ \ell_{2}\ contains vectors that are not finitely supported. Therefore, \ H \eq F \, and because \(F^{\perp} \) contains only the zero vector, \( H \eq F + F^{\perp}\).

Highlight the Importance of Closedness

This example demonstrates the importance of the closedness assumption. \ F \ is not closed in \ \ell_{2}\. If \(F \) were closed, then \(H = F + F^{\perp}\) would hold. The closedness ensures that every element is adequately represented.

Key concepts

Orthogonal Complement

When working with Hilbert spaces, the **orthogonal complement** plays a crucial role. The orthogonal complement of a subspace \( F \) in a Hilbert space \( H \) is denoted as \( F^{\perp} \). It consists of all vectors in \( H \) that are orthogonal to every vector in \( F \). Mathematically, this can be expressed as:
e.g., \( F^{\perp} = \{ y \in H \mid \, \langle y, x \rangle = 0 \,\forall x \in F \} \)
In our example, the orthogonal complement of the subspace of finitely supported vectors in \( \ell_{2} \) is the zero vector \( \{0\} \). This is because any non-zero vector in \( \ell_{2} \), a space of square-summable sequences, cannot be orthogonal to every finitely supported vector.
Understanding orthogonal complements helps in visualizing and decomposing Hilbert spaces into simpler, non-overlapping parts.

Closed Subspaces

Closed subspaces are fundamental in Hilbert spaces. A subspace \( F \) in a Hilbert space \( H \) is closed if it contains all its limit points. In other words, if a sequence of vectors in \( F \) converges to a vector in \( H \), then this limit vector is also in \( F \).

Consider the subspace of finitely supported vectors in \( \ell_{2} \). This subspace, \( F \), is not closed. To see why, note that sequences of finitely supported vectors can converge to a vector that is not finitely supported. Therefore, assuming closedness is necessary for certain theorems concerning orthogonal complements and subspace decomposition.
The example from the exercise highlights the importance of closedness by showing that if \( F \) were closed, then the space \( H \) would decompose into \( F \) and its orthogonal complement \( F^{\perp} \). Since \( F \) is not closed, this decomposition fails.

Square-Summable Sequences

Square-summable sequences are a key concept in the study of Hilbert spaces, particularly \( \ell_{2} \). A sequence \( x = (x_{1}, x_{2}, x_{3}, \ldots) \) in \( \ell_{2} \) is square-summable if the sum of the squares of its elements is finite, formally expressed as:
e.g., \( \sum_{i=1}^{\infty} | x_{i} |^{2} < \infty \)
This property ensures that sequences in \( \ell_{2} \) have finite norm, making it a complete and separable space.
Square-summable sequences are essential in various applications, including signal processing and quantum mechanics. They allow for the decomposition of functions into simpler, orthogonal components, making analysis and computation more feasible.
In our example, \( \ell_{2} \) contains vectors that are not finitely supported, which demonstrates that the subspace \( F \) of finitely supported vectors does not encompass all of \( \ell_{2} \), emphasizing the difference between finitely supported and square-summable sequences.