Question

Let \(H\) be an infinite-dimensional separable Hilbert space. Show that \(H\) admits a norm that is not equivalent to the original norm. Hint: If \(\left\\{e_{i}\right\\}_{i=1}^{\infty}\) is an orthonormal basis of \(H\), put \(\|x\|=\sum 2^{-i}\left|x_{i}\right|\) for \(x=\sum x_{i} e_{i} .\) Check this norm on \(\left\\{e_{i}\right\\}\).

Short answer

The new norm \( |x| = \sum_{i=1}^{\infty} 2^{-i} |x_i| \) is not equivalent to the original norm since it scales down the basis vectors geometrically.

Step-by-step solution

Identify the Orthonormal Basis

Start by identifying an orthonormal basis \(\{e_i\}\) for the infinite-dimensional separable Hilbert space \(H\). Every vector \(x\) in the space can be written as \(x = \sum_{i=1}^{\infty} x_i e_i\).

Define the New Norm

Define a new norm for the Hilbert space using the series \(|x| = \sum_{i=1}^{\infty} 2^{-i} |x_i|\). This norm is based on the coefficients of \(x\) with respect to the orthonormal basis.

Check the New Norm on Basis Vectors

Check the norm on the basis vectors \(e_i\). For each basis vector \(e_i\), evaluate \(|e_i|\). Since \(e_i\) has a coefficient of \(1\) in the \(i\)-th position and \(0\) elsewhere, \(|e_i| = 2^{-i}\). Hence, \(|e_i|\) diminishes geometrically with \(i\).

Compare the Original and New Norms

Compare the original norm and the new norm. The original norm of each \(e_i\) is \(||e_i|| = 1\), while the new norm \(|e_i| = 2^{-i}\) does not remain consistent with the original norm.

Conclusively Show Non-Equivalence

To conclusively show that the norms are not equivalent, note that the new norm \(|\cdot|\) applied to the sequences of basis vectors scales down geometrically, whereas the original norm \(||\cdot||\) is constant. Therefore, there does not exist a constant \(C > 0\) such that \(C^{-1}||x|| \leq |x| \leq C||x||\) for every \(x \in H\).

Key concepts

Infinite-Dimensional Hilbert Space

A Hilbert space is a complete inner-product space, which can be finite or infinite-dimensional. In infinite-dimensional Hilbert spaces, we often deal with sequences rather than finite sums of vectors. One of the crucial properties is that every vector in such a space can be represented as a series with respect to an orthonormal basis. Think of it like representing any point in the space as a combination of infinitely many basis elements. For example, in the given exercise, the space is denoted by \(H\) and is separable, meaning it has a countable orthonormal basis.

Orthonormal Basis

An orthonormal basis in a Hilbert space is a set of vectors that are orthogonal (perpendicular) to each other and each have a norm of one. This means they are unit vectors. In our context, an orthonormal basis \(\{e_i\}_{i=1}^{\infty}\) for \(H\) allows us to write any vector \(x\) in \(H\) as \(x = \sum_{i=1}^{\infty} x_i e_i\), where \(x_i\) are the coefficients, also known as the Fourier coefficients, with respect to this basis. This decomposition helps simplify complex problems in Hilbert space into potentially more manageable series forms.

Norm Equivalence

Two norms \(||\bullet||\) and \(|\bullet|\) on a vector space are said to be equivalent if there exist positive constants \(C_1, C_2 > 0\) such that for all vectors \(x\) in the space, \(C_1 ||x|| \le |x| \le C_2 ||x||\). This means, the norms scale each other uniformly across the space. In the given exercise, we prove that the original norm \(||\bullet||\) and the new norm defined by \(|x| = \sum_{i=1}^{\infty} 2^{-i} |x_i|\) are not equivalent because the scaling factor varies for each basis vector \(e_i\) as \(2^{-i}\), which diminishes as \(i\) increases, thus failing the uniform scaling condition.

Functional Analysis

Functional analysis is the study of vector spaces with infinite dimensions and the linear operators acting upon them. It uses concepts like norms, inner products, and orthonormal bases to understand the structure and behavior of functions, which can be infinite sequences in spaces like \(H\). In this field, showing that a space can have different norms that are not equivalent helps in understanding the diversity of possible structures and behaviors within these spaces. This is what we establish in the given exercise, illustrating how choosing different norms can lead to fundamentally different scaling and measuring processes within the same space.