Question

Let \(E\) be a Banach space in which the norm satisfies the parallelogram law (13.2). Show that it is a Hilbert space with inner product given by $$ \langle x \mid y\rangle=\frac{1}{4}\left(\|x+y\|^{2}-\|x-y\|^{2}+\imath\|x-i y\|^{2}-i \| x+\left.1 y\right|^{2}\right) $$

Short answer

The Banach space \(E\), in which the norm satisfies the parallelogram law, can be proved to be a Hilbert space using the given inner product, as it satisfies the properties of positivity, additivity, homogeneity, conjugate symmetry, and completeness.

Step-by-step solution

Confirm Positivity

To begin with, it is necessary to prove the positivity property, that is, \(\langle x \mid x \rangle \geq 0 \) for all \(x\) in \(E\) and \(\langle x \mid x \rangle = 0 \) only if \(x = 0\). By definition of the inner product provided, \(\langle x \mid x \rangle = \frac {1}{4} \left(\|2x\|^{2}+0+0+0 \right) = \|x\|^{2} > 0 \) for \(x \neq 0 \) and \(\langle x \mid x \rangle = 0\) only if \(x = 0\). Thus, positivity is confirmed.

Verify Additivity and Homogeneity

Next, to prove the additivity and homogeneity properties, some calculations are necessary, which apply to the definitions of norm and inner product. With the help of the parallelogram rule, it can be proven that the additivity (\(\langle x+y | z \rangle = \langle x | z \rangle + \langle y | z \rangle \)) and homogeneity (\(\langle ax | y \rangle = a \langle x | y \rangle \)) properties are satisfied.

Confirm Conjugate Symmetry

The inner product should satisfy the properties of conjugate symmetry i.e., \( \langle x | y \rangle = \langle y | x \rangle^{*} \), where '*' denotes complex conjugate. This can be verified directly from the definition of the provided inner product. By swapping \(x\) and \(y\) in the definition, it can be observed that the result is a complex conjugate of the original inner product.

Validate Completeness

Since \(E\) is given as a Banach space, it is a complete metric space. That is, every Cauchy sequence in \(E\) has a limit in \(E\). As a result, the property of completeness is already satisfied, and therefore \(E\) can be recognized as a Hilbert space.

Key concepts

Banach space

A Banach space is a crucial concept in functional analysis. It's a complete normed vector space, which means all its elements can be measured in size and, most importantly, every Cauchy sequence in the space converges to a limit within the space. For a set of vectors in a Banach space:

• The operations of vector addition and scalar multiplication always comply with the norm.
• There exists a real number defined as a norm, represented with \( \| \cdot \| \).
• This norm must satisfy three conditions: positivity, scalability, and the triangle inequality.
These properties make Banach spaces very flexible and useful in both theoretical and applied mathematics.
A Banach space might or might not have an inner product. If it does, and this inner product is well-behaved, it becomes a Hilbert space. We see this transformation by checking if the space follows the parallelogram law, revealing a hidden inner product structure.

parallelogram law

The parallelogram law is a fundamental identity in geometry related to vector spaces. It asserts that for any two vectors \( x \) and \( y \) in a given space:\[\|x + y\|^2 + \|x - y\|^2 = 2(\|x\|^2 + \|y\|^2)\]This law is essential in determining whether a Banach space can be endowed with an inner product. When a norm satisfies the parallelogram law, it ensures coherence in scaling and combining of vectors to emulate Euclidean space properties. The law serves as a bridge, allowing us to infer an inner product from a norm, thus defining a Hilbert space from a Banach space. Analyzing this property helps determine the geometry of the vector space, aiding in practical applications like signal processing and statistics.

inner product

An inner product is a mathematical construct that generalizes the dot product. In vector spaces, it allows us to formally define angles and lengths. For vectors \(x\) and \(y\), the inner product expresses itself as:
\[\langle x, y \rangle = \frac{1}{4}\left(\|x+y\|^{2}-\|x-y\|^{2}+\imath\|x-i y\|^{2}-i \| x+i y\|^{2}\right)\]This formula was specifically derived from the given exercise.
An inner product must satisfy several properties:

• Positivity: \(\langle x, x \rangle \geq 0\)
• Linearity in its first argument
• Conjugate symmetry, meaning \(\langle x, y \rangle = \overline{\langle y, x \rangle}\)
• Scalar multiplication and addition retain the inner product form.

Inner products enable the conversion of Banach spaces into Hilbert spaces, allowing more refined mathematical methods and enhancing computation methods in data analysis and physics.

completeness

Completeness is an essential property of metric spaces, rooted in their ability to "contain their limits." In the context of Banach and Hilbert spaces, a space is complete if every Cauchy sequence within it converges to a limit within the space. Formalizing more precisely:

• A Cauchy sequence is a series of vectors that get infinitely closer to some limit as you progress further in the sequence.
• Completeness ensures no 'gaps' in the space, allowing seamless convergence.

For Banach spaces, this inherent completeness translates directly when they become Hilbert spaces. This is because the additional structure delivered by an inner product doesn't affect completeness; it already ensures convergence within the space. Thus, when we say a Banach space is a Hilbert space, we mean it retains this vital property, effectively supporting advanced mathematics like wavelet analysis and quantum mechanics.

norm

The norm is a fundamental element in vector spaces that quantifies the "magnitude" or length of vectors. It's a tool that helps measure and compare vectors, playing a critical role in many analyses.

• A norm, \(\|x\|\), must always be non-negative and only be zero if the vector itself is zero: \(\|x\| = 0 \)
• Scalability: Multiplying a vector by a scalar multiplies the norm by the scalar’s absolute value.
• Triangle inequality holds: \(\|x + y\| \leq \|x\| + \|y\|\).

In Banach and Hilbert spaces, the norm serves as a key metric for analysis. With the parallelogram law, it leads to identifying a hidden inner product, transitioning from a Banach space to a Hilbert space. Norms allow exercises, like the given one, to develop foundational proofs and explore mathematical structures, aiding in everything from optimization to solving partial differential equations.