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Chapter 13: Problem 1

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

Expert verified
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

01

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.
02

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.
03

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.
04

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.

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Most popular questions from this chapter

If \(A_{1}, A_{2}, \ldots . A_{n}\) are operators on a dense domain such that $$ \sum_{i=1}^{n} A_{1}^{*} A_{1}=0 $$ show that \(A_{1}=A_{2}=\cdots=A_{n}=0 .\)

Which of the following is a vector subspace of \(\ell^{2}\), and which are closed? In each case find the space of vectors orthogonal to the set. (a) \(V_{N}=\left\\{\left(x_{1}, x_{2} \ldots\right) \in \ell^{2} \mid x_{1}=0\right.\) for \(\left.i>N\right]\) (b) \(V=\bigcup_{N=1}^{\infty} V_{N}=\left|\left(x_{1}, x_{2}, \ldots\right) \in \ell^{2}\right| x_{t}=0\) for \(i>\) some \(\left.N\right]\). (c) \(U=\left|\left(x_{1}, x_{2}, \ldots\right) \in \ell^{2}\right| x_{1}=0\) for \(\left.t=2 n\right\\} .\) (d) \(W=\left\\{\left(x_{1}, x_{2}, \ldots\right) \in \ell^{2} \mid x_{1}=0\right.\) for some \(\left.i\right\\}\)

Show that the sum of two projection operators \(P_{u}+P_{M}\) is a projection operator iff \(P_{M} P_{N}=0\). Show that this condition is equavalent to \(M \perp N\).

Show that if \(\left(A, D_{A}\right)\) and \(\left(B, D_{B}\right)\) arc operators on dense domains in \(H\) then \(B^{*} A^{*} \subseteq\) \((A B)^{\circ}\)

Verify that the operator on three-dimensonal Hilbert space, having matrix representation in an o.n. basis $$ \left(\begin{array}{ccc} \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 1 & 0 \\ -\frac{1}{2} & 0 & \frac{1}{2} \end{array}\right) $$ is a projection operator, and find a basis of the subspace it projects onto.
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