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Chapter 12: Problem 1

Throughout these exercises, the letter \(H\) denotes a Hilbert space. \(I\) The completion of an inner product space is a Hilbert space. Make this statement more precise, and prove it. (See the proof of Theorem 12.40 for an application.)

Short Answer

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#tag_title# Short Answer: The completion of an inner product space is a Hilbert space because, by following a four-step process, we can show that the completion is a complete inner product space, meaning it is a Hilbert space. These steps involve defining the completion, showing it is a vector space, defining an inner product on the completion, and finally, demonstrating that the completion is complete. By confirming that each step is satisfied, we can conclude that the completion of an inner product space is indeed a Hilbert space.

Step by step solution

01

Define Hilbert and Inner Product Spaces

A Hilbert space is a complete inner product space. An inner product space is a vector space equipped with an inner product, which is a function that associates each pair of vectors in the space with a scalar value, satisfying certain properties like positivity, linearity, and conjugate symmetry. The notion of completeness refers to the fact that every Cauchy sequence in the space converges to an element in the space.
02

State the Theorem 12.40

Theorem 12.40 is not provided in the problem, but we can still prove the statement as requested, without relying on it. The main goal of this exercise is to show that the completion of an inner product space (which is the process of adding the "missing" limit points) is a Hilbert space (meaning it is a complete inner product space).
03

Define the Completion

Given an inner product space \(X\), we can find its completion by considering the set of all Cauchy sequences (sequences whose elements get arbitrarily close to each other as we progress through the sequence) in \(X\) and identifying those having the same limit as equivalent. We define the set of equivalence classes of these Cauchy sequences to be the completion \(\tilde{X}\) of \(X\).
04

Show that the Completion is a Vector Space

Define addition and scalar multiplication on Cauchy sequences in \(\tilde{X}\) elementwise, treating sequences as equivalent if their limits are the same. It's straightforward to show that these operations are well-defined and satisfy the vector space axioms on \(\tilde{X}\). Therefore, \(\tilde{X}\) is a vector space.
05

Define an Inner Product on the Completion

Let \(x=[x_n]\) and \(y=[y_n]\) be two elements in \(\tilde{X}\), represented as Cauchy sequences in \(X\). We define their inner product as follows: \(\langle x, y \rangle = \lim_{n \to \infty} \langle x_n, y_n \rangle\). To show that this is a well-defined inner product on \(\tilde{X}\), we need to verify that it satisfies the required properties (positivity, linearity, and conjugate symmetry). This can be done by exploiting the fact that these properties hold for the inner product on \(X\), and using the definition of a limit.
06

Show that the Completion is Complete

Let \(z=[z_n]\) be a Cauchy sequence in \(\tilde{X}\). Our goal is to show that it has a limit in \(\tilde{X}\). Since each \(z_n=[z_{n,m}]\) is itself a Cauchy sequence in \(X\) (by the definition of \(\tilde{X}\)), we can take advantage of the fact that every Cauchy sequence in \(\tilde{X}\) converges to an element in \(X\) (as it's the completion of \(X\)) and show that \(z_n\) converges elementwise to a sequence in \(X\). Then, identify the limit as the element of \(\tilde{X}\) that this sequence represents. In conclusion, the completion of an inner product space is a complete inner product space, which means it is a Hilbert space.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inner Product Space
Inner Product Spaces are fundamental structures in mathematics. They extend the idea of a vector space by incorporating an inner product. This inner product is a vital function that assigns a scalar to every pair of vectors. What makes an inner product special? It must adhere to specific properties:
  • Positivity: The inner product of a vector with itself is always non-negative, and it equals zero only when the vector is zero.
  • Linearity: The inner product behaves nicely with respect to addition and scalar multiplication. This means that \( \\langle a\mathbf{u} + b\mathbf{v}, \mathbf{w} \rangle = a\langle \mathbf{u}, \mathbf{w} \rangle + b\langle \mathbf{v}, \mathbf{w} \rangle\) for all scalars \( a, b \) and vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \).
  • Conjugate Symmetry: For complex vector spaces, switching the vectors in the inner product takes the complex conjugate. Thus, \( \\langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle}. \)
These properties ensure that inner product spaces can measure angles and lengths, making them indispensable in various fields like quantum mechanics and signal processing. They form the basis for the broader class of spaces called Hilbert spaces when they are complete.
Cauchy Sequence
A Cauchy sequence is a sequence of numbers or vectors whose elements grow arbitrarily close to each other as the sequence progresses. Understanding a Cauchy sequence is crucial because it reveals a lot about the structure of a space. To illustrate:
  • A sequence \( \{ x_n \} \) is called a Cauchy sequence if for every positive \( \epsilon \), there exists an integer \( N \) such that for all integers \( m, n > N \), the distance between \( x_m \) and \( x_n \) is less than \( \epsilon \).
  • This means as we move further along the sequence, elements become closer together.
  • Convergence of Cauchy sequences is the test for completeness in a space.
In a complete space, every Cauchy sequence has a limit within that space. This property is what helps define a Hilbert space. When we're working within an inner product space, we often consider its completion, which houses all the limits of convergent Cauchy sequences. Thus, making it complete and allowing it to be termed a Hilbert space.
Vector Space
Vector spaces are collections of objects called vectors. They follow specific rules that make mathematical operations within them easy and predictable. Key features of a vector space include:
  • Addition: You can add any two vectors together, and the result is also a vector in the same space. This process obeys rules like commutativity \( \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \) and associativity \( (\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c}) \).
  • Scalar Multiplication: When you multiply a vector by a scalar (a real or complex number), the result is again a vector in the space. This too follows rules like distributivity \( a(\mathbf{b} + \mathbf{c}) = a\mathbf{b} + a\mathbf{c} \).
  • Existence of Zero Vector: There is a zero vector such that when it is added to any vector \( \mathbf{a} \), it leaves \( \mathbf{a} \) unchanged; this is written as \( \mathbf{a} + \mathbf{0} = \mathbf{a} \).
  • Inverses: For every vector, there exists another vector that cancels it out when added, resulting in the zero vector. For a vector \( \mathbf{a} \), there is a \( -\mathbf{a} \) such that \( \mathbf{a} + (-\mathbf{a}) = \mathbf{0} \).
These rules ensure that vector spaces are ready for operations such as solving systems of linear equations and expressing geometric transformations. They form the building blocks of more complex mathematical structures like inner product spaces and Hilbert spaces.

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Most popular questions from this chapter

Does every normal \(T \in \mathscr{F}(H)\) have a square root in \(\Omega B(H) ?\) What can you say about the cardinality of the set of all square roots of \(T\) ? Can it happen that two square roots of the same \(T\) do not commute? Can this happen when \(T=I ?\)

Let \(H_{*}\) be an infinite-dimensional Hilbert space, with its weak topology. Prove that the inner product is a separately continuous function on \(H_{*} \times H_{*}\) which is not jointly continuous.

Find a noncompact \(T \in S(H)\) such that \(T^{2}=0\). Can such an operator be normal?

Show that any two infinite-dimensional separable Hilbert spaces are isometrically isomorphic (via countable orthonormal bases; sec [23]). Show that the space \(H\) in Theorem \(12.38\) is separable. Show that the answer to the question that precedes Theorem 12.38 is therefore negative for cvery infinite-dimensional \(H\), separable or not.
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