Question

The norm \(\|\phi\|\) of a bounded linear operator \(\phi: \mathcal{H} \rightarrow \mathrm{C}\) is defined as the greatest lower bound of all \(M\) such that \(|\phi(u)| \leq M\|u\|\) for all \(u \in \mathcal{H}\). If \(\phi(u)=(v \mid u)\) show that \(\|\phi\|=\|v\|\). Hence show that ahe bounded lanear functional norm satisfies the parallelogram law $$ \|\phi+\vartheta\|^{2}+\|\phi-\psi\|^{2}=2\|\phi\|^{2}+2\|\psi\|^{2} $$

Short answer

The norm of the bounded linear operator \(\phi\), given by \(\|\phi\|=\|v\|\), has been proved successfully by leveraging the Cauchy-Schwarz inequality. Moreover, the parallelogram law for the norm of a bounded linear functional holds. Hence, the respective equalities are \(\|\phi\| = \|v\|\) and \(\|\phi+\vartheta\|^{2}+\|\phi-\psi\|^{2}=2\|\phi\|^{2}+2\|\psi\|^{2}\).

Step-by-step solution

Proving the Equality of Norms

First, it is needed to show that \(\|\phi\|=\|v\|\). Given that \(|\phi(u)|=(v|u)\), by applying the Cauchy-Schwarz inequality, it can be found that \(|\phi(u)| \leq \|v\|\|u\|\) for all \(u \in \mathcal{H}\). This implies \(\|\phi\| \leq \|v\|\). In order to complete the proof, the opposite direction of the inequality must be shown. Choosing \(u = \dfrac{v}{\|v\|}\) and substituting it into the equation of the operator \(\phi\), it can be seen that \(|\phi(u)| = \|v\|\). Hence, \(\|\phi\|=\|v\|\).

Applying the Parallelogram Law

Knowing that \(\|\phi\|=\|v\|\), the parallelogram law can now be applied to the vector \(v\), which should hold since \(\mathcal{H}\) is assumed to be a Hilbert space. For general vectors \(v\) and \(w\), the parallelogram law is given by \(\|v+w\|^2+\|v-w\|^2=2\|v\|^2+2\|w\|^2\). Substituting the result from step 1 into the parallelogram law, the proof that the norm of the bounded linear functional á satisfies the parallelogram law should hold, which is given by: \(\|\phi+\vartheta\|^{2}+\|\phi-\psi\|^{2}=2\|\phi\|^{2}+2\|\psi\|^{2}\)

Key concepts

Bounded Linear Operators

A bounded linear operator is a fundamental concept in functional analysis, often encountered when dealing with linear mappings between different spaces. It is defined as a linear operator \( T \) such that, for all vectors \( u \) in a Hilbert space \( \mathcal{H} \, \), there exists a constant \( M \) satisfying: \(|Tu| \leq M|u|\). The smallest such \( M \) provides the operator norm, formally completing the definition of a bounded operator.

Bounded linear operators ensure that the operator does not "blow up" on the space; it maintains a sense of control over the transformation of vectors. This kind of stability is crucial when analyzing operator behavior, as it allows us to ensure that the output remains within predictable limits.

In many practical applications, bounded linear operators play a role in simplifying complex problems by allowing the use of easily manipulated linear transformations.

• They maintain continuity, critical for theoretical and practical considerations.
• They facilitate the extension of finite-dimensional linear algebra concepts to infinite-dimensional spaces.

Parallelogram Law

The parallelogram law is an identity often used in normed vector spaces, especially in Hilbert spaces. It asserts that for any vectors \( u \) and \( v \) in the space, the following equation holds: \(|u+v|^2 + |u-v|^2 = 2|u|^2 + 2|v|^2\).

This law visually represents the way vector norms add together in a space, and it is named so because it relates to the geometry of parallelograms. The sum of the squares of the diagonals (\(|u+v|^2\) and \(|u-v|^2\)) is equal to the sum of the squares of all sides (multiplied by two).

This principle is not only a neat geometrical fact but also a characteristic property of inner product spaces. Inner product spaces that comply with the parallelogram law can be classified as Hilbert spaces. This connection is essential for confirming the structure of the space and ensuring that further operations like projections or orthogonal decompositions can be accurately performed.

It also plays a pivotal role in the proof of various properties of norms and is frequently employed in the analysis and simplification of problems involving bounded linear operators.

Norms in Functional Analysis

Norms serve as a fundamental tool in functional analysis, providing a way to measure sizes or lengths of vectors in various spaces. A norm is a function \( \| \cdot \| \) that assigns a non-negative scalar to each vector in the space, adhering to certain axioms such as positivity, scalar multiplication, and the triangle inequality.

In the context of a Hilbert space, the norm is induced by the inner product, which induces a structure called the "normed space." The inner product \( (u|v) \) leads to the natural definition of a norm \( \|u\| = \sqrt{(u|u)} \). This specific norm highlights the geometric properties of the space, allowing for operations such as distance measurement and projection, making it indispensable.

In applications, norms help:

• To ensure stability and controllability of operators.
• To facilitate error estimation in approximations and numerical methods.
• To enable comparison and convergence analysis within the space.

These aspects are crucial in handling infinite-dimensional spaces and provide a grounding for rigorous mathematical proofs.

Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality is a cornerstone result in both algebra and analysis, particularly within the context of inner product spaces. It explains the relationship between the inner product and the norms of two vectors. Formally, for vectors \( u \) and \( v \) in a Hilbert space, it states: \( |(u|v)| \le \|u\| \|v\|\).

This inequality implies that the absolute value of the inner product is always less than or equal to the product of the norms of the vectors. This has significant implications in maintaining the consistency and interrelation of measurements within the space, ensuring that no vector "outgrows" its set boundaries in the space due to manipulations provided by inner products.

The Cauchy-Schwarz inequality serves various purposes, including:

• Proving other crucial inequalities and identities within analysis.
• Ensuring the orthogonality of vectors is correctly adhered to and maintained.
• Providing the foundation for projections and angle analysis between vectors.

In the exercise context, it was crucial for showing that \( |\phi(u)| = (v|u) \le \|v\| \|u\|\), forming the groundwork for understanding the operator norms.