Question

Let }\left\\{e_{i}\right\\} \text { be an orthonormal basis of a Hilbert space } H . \text { Show that }\\\ &e_{i} \stackrel{w}{\rightarrow} 0 \end{aligned} $$

Short answer

The orthonormal basis \( e_{i} \stackrel{w}{\rightarrow} 0 \) because the inner product \( \langle e_{i}, y \rangle \rightarrow 0 \) for all \( y \) in the Hilbert space.

Step-by-step solution

Title - Understand Basis and Weak Convergence

An orthonormal basis \( \{e_{i}\} \) of a Hilbert space H is a set of vectors that are mutually orthogonal and each vector has unit norm. The notation \( e_{i} \stackrel{w}{\rightarrow} 0 \) means that \( e_{i} \) converges to 0 in the weak topology.

Title - Properties of Orthonormal Basis

Recall that for any vector \( x \) in the Hilbert space H, the orthonormal basis \( \{e_{i}\} \) satisfies \( \langle e_{i}, e_{j} \rangle = \delta_{ij} \), where \( \delta_{ij} \) is the Kronecker delta function.

Title - Define Weak Convergence

A sequence \( x_i \) in a Hilbert space converges weakly to a vector x if for all \( y \) in the Hilbert space H, \( \langle x_i, y \rangle \rightarrow \langle x, y \rangle \). In this case, we want to show that \( \langle e_{i}, y \rangle \rightarrow \langle 0, y \rangle = 0 \) for all y.

Title - Applying the Property to Orthogonal Basis

For any \( y \) in the Hilbert space H, \( y \) can be expressed as \( y = \sum_{k} \langle y, e_{k} \rangle e_{k} \). Now consider the inner product \( \langle e_{i}, y \rangle. \) Since \( e_{i} \) and \( e_{k} \) are orthogonal, \( \langle e_{i}, y \rangle = \langle e_{i}, \sum_{k} \langle y, e_{k} \rangle e_{k} \rangle = \langle y, e_{i} \rangle. \)

Title - Show Convergence to Zero

Since \( \{e_{i}\} \) is an orthonormal basis, \( e_{i} \) is orthogonal to almost all elements of the basis \( e_{k} \). This means \( \langle e_{i}, y \rangle = 0 \) when \( i eq k \. \) As \( i \rightarrow \infty \), \( \langle e_{i}, y \rangle = \langle y, e_{i} \rangle \rightarrow 0 \), confirming weak convergence to zero.

Key concepts

orthonormal basis

An orthonormal basis in a Hilbert space is very important.
It consists of vectors that are both orthogonal and of unit length.
Orthogonal means that the inner product of any two distinct basis vectors is zero.
Unit length means the inner product of any basis vector with itself is one.

For example, in a two-dimensional Euclidean space, the standard unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) form an orthonormal basis:
\[ \mathbf{i} = (1, 0), \mathbf{j} = (0, 1) \] Here, \( \langle \mathbf{i}, \mathbf{j} \rangle = 0 \, \langle \mathbf{i}, \mathbf{i} \rangle = 1 \, \langle \mathbf{j}, \mathbf{j} \rangle = 1 \)
This property makes calculations easier and helps us understand the structure of Hilbert spaces.
Using an orthonormal basis, any vector in the Hilbert space can be expressed as a sum of these basis vectors.

Hilbert space

A Hilbert space is a complete vector space with an inner product.
Completeness means every Cauchy sequence in the space has a limit within the space.
This property makes Hilbert spaces very useful in mathematics and physics.
The inner product is a way of multiplying two vectors to get a scalar.
For vectors \(x, y \) in a Hilbert space, the inner product is usually denoted \( \langle x, y \rangle \).
This leads to concepts like norm and orthogonality.

The norm of a vector \( x \) in a Hilbert space is given by \( \sqrt{ \langle x, x \rangle } \)
Orthogonality uses the inner product to determine when two vectors are perpendicular:
\[ \langle x, y \rangle = 0 \Rightarrow x \text{ and } y \text{ are orthogonal} \] Hilbert spaces are used in many areas like quantum mechanics and signal processing.
They help solve problems involving infinite-dimensional spaces.

inner product

The inner product is a key concept in Hilbert spaces.
It pairs two vectors and results in a scalar, providing a notion of angle and length.
The inner product has three main properties:

• Conjugate Symmetry: \ \langle x, y \rangle = \overline{ \langle y, x \rangle } \
• Linearity in the first slot: \ \langle ax+by, z \rangle = a \langle x, z \rangle + b \langle y, z \rangle \
• Positive-Definiteness: \ \langle x, x \rangle \geq 0 \text{ and } \langle x, x \rangle = 0 \text{ iff } x = 0 \

This makes the inner product a generalization of the dot product in Euclidean spaces.

For example, in \( \mathbb{R}^n \), the inner product of two vectors \( x \) and \( y \) is:
\[ \langle x, y \rangle = \sum_{i=1}^{n} x_i y_i \] This can also extend to complex numbers and functions.
The inner product allows us to measure distances and angles in more complex spaces.

Kronecker delta

The Kronecker delta function is a special function denoted \( \delta_{ij} \).
It takes two indices and returns 1 if they are equal and 0 otherwise:
\[ \delta_{ij} = \begin{cases} 1 & \text{if } i = j \ 0 & \text{if } i eq j \end{cases} \] The Kronecker delta is crucial in simplifying sums and expressions involving orthonormal bases.
If \( \{ e_i \} \) is an orthonormal basis in a Hilbert space:
\[ \langle e_i, e_j \rangle = \delta_{ij} \] This property ensures that the basis vectors are mutually orthogonal and have unit length.

It's commonly used in both finite and infinite-dimensional spaces.
The Kronecker delta helps in efficiently working with sums and simplifies many problems.

weak topology

Weak topology is an important concept when working with infinite-dimensional spaces.
In weak topology, a sequence \( x_i \) in a Hilbert space converges weakly to \( x \) if:
\[ \langle x_i, y \rangle \rightarrow \langle x, y \rangle \text{ for all } y \in H \] This is weaker than strong convergence, where \( x_i \rightarrow x \) directly.
In practice, this means the inner products with any other vector approach the expected values.

For instance, if \( e_i \) is part of an orthonormal basis:
\[ e_i \stackrel{w}{\rightarrow} 0 \text{ if } \langle e_i, y \rangle \rightarrow 0 \text{ for all } y \in H \] Weak convergence is useful in many mathematical areas, including functional analysis.
It allows for flexibility and aids in handling more complex infinite-dimensional problems.