Problem 10 - A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry

Question

If \(S\) is any subset of \(\mathcal{H}\), and \(V\) the closed subspace generated by \(S, V=\overline{L(S)}\), show that \(S^{1}=\left\\{u \in \mathcal{H} \mid\\{u|x\rangle=0\right.\) for all \(x \in S\\}=V^{1}\)

Step-by-Step Solution

Answer

The orthocomplement of \(S\), denoted by \(S^{1}\), is in fact equal to the orthocomplement of \(V\), \(V^{1}\), where \(V\) is the closed subspace generated by \(S\). This is shown by first proving that \(S^{1} \subseteq V^{1}\) and then that \(V^{1} \subseteq S^{1}\), which can then be used to conclude that \(S^{1} = V^{1}\).

Step 1: Show that \(S^{1} \subseteq V^{1}\)

Start by assuming that \(u \in S^{1}\). This means that for every \(x \in S\), \(u, x = 0\), where ',' represents the inner product. Hence, \langle u, x \rangle = 0 for all \(x \in S\). Because every \(x \in S\) is also in \(V\) (since \(V\) is generated by \(S\)), then \langle u, x \rangle = 0 holds for all \(x \in V\). Therefore, \(u \in V^{1}\), effectively showing that \(S^{1} \subseteq V^{1}\).

Step 2: Show that \(V^{1} \subseteq S^{1}\)

Again, start by assuming that \(u \in V^{1}\). This means that \langle u, x \rangle = 0 for all \(x \in V\). Now take a sequence \(x_n \in L(S)\) such that \(x_n\) converges to \(x\) in \(V\). Since \(x_n \rightarrow x\), and \(u \in V^{1}\) (meaning that \langle u, x_n \rangle -> 0), it must also be the case that \langle u, x \rangle = 0 for all \(x \in V\), meaning that \(u \in S^{1}\). This shows that \(V^{1} \subseteq S^{1}\).

Step 3: Conclude that \(S^{1} = V^{1}\)

Having shown in steps 1 and 2 that \(S^{1} \subseteq V^{1}\) and \(V^{1} \subseteq S^{1}\), it can now be concluded that \(S^{1} = V^{1}\), which is the outcome the exercise requested.

Key Concepts

Subspace

In the context of Hilbert spaces, a **subspace** is a subset that retains the structure of the larger space. For any subset \( S \) in a Hilbert space \( \mathcal{H} \), the subspace \( V \) generated by \( S \) is crucial. This means \( V \) includes all possible linear combinations of the elements in \( S \), and if \( V \) is closed, it also contains all limits of convergent sequences in \( S \).
For example:

• The set of all vectors in \( \mathcal{H} \) that can be formed as \( a_1 x_1 + a_2 x_2 + \ldots + a_n x_n \) where \( x_i \in S \) is part of the subspace.
• If \( V \) is closed, then even if a sequence of vectors from \( S \) converges to another vector, that vector is also in \( V \).

This property makes \( V \) a powerful concept in analyzing the relationships between different parts of a Hilbert space.

Inner Product

The **inner product** is a special operation that takes two vectors from a Hilbert space and returns a scalar. It helps to define geometric concepts like angle and length within abstract spaces.
Consider two vectors \( u \) and \( v \) in a Hilbert space \( \mathcal{H} \). The inner product is denoted as \( \langle u, v \rangle \) and must satisfy several properties:

• Linearity: \( \langle au + bw, v \rangle = a \langle u, v \rangle + b \langle w, v \rangle \)
• Symmetry: \( \langle u, v \rangle = \overline{\langle v, u \rangle} \)
• Positivity: \( \langle u, u \rangle \ge 0 \) and equals zero only if \( u \) is the zero vector.

These properties make the inner product a cornerstone in understanding the distance and angles between vectors, forming the basis for **orthogonality**.

Orthogonality

**Orthogonality** refers to the concept of vectors being perpendicular to each other, which is central in Hilbert space theory. Two vectors \( u \) and \( v \) are orthogonal if their inner product is zero, i.e., \( \langle u, v \rangle = 0 \).
In subspaces:

• If every vector \( u \) in subspace \( V^{\perp} \) (the orthogonal complement) is orthogonal to every vector in \( V \), then \( V^{\perp} \) contains all vectors in \( \mathcal{H} \) orthogonal to \( V \).
• This concept helps decompose spaces into simpler, independent elements, enhancing the ability to analyze complex structures within Hilbert spaces.

Orthogonality simplifies calculations and is widely used in signal processing and quantum mechanics, where separating components into orthogonal parts can make problems far more manageable.

Convergence

**Convergence** in a Hilbert space involves a sequence of vectors approaching a specific vector in terms of distance. This concept is vital for understanding limits and continuity within these spaces.
A sequence \( \{ x_n \} \) is said to converge to \( x \) if for any small positive value \( \epsilon \), there exists an integer \( N \) such that for all \( n > N \), the distance \( \| x_n - x \| < \epsilon \).
This means:

• As \( n \) becomes larger, \( x_n \) gets arbitrarily close to \( x \).
• Convergence ensures stability and predictability of sequences, allowing us to effectively work within subspaces like \( V \).

In our specific problem, the concept of convergence helps relate sequences in \( S \) to the closed subspace \( V \), providing a bridge to understanding how different vectors interact within \( \mathcal{H} \).