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Chapter 13: Problem 11

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 \(\left.i\right\\}\)

Short Answer

Expert verified
The sets \(V_N\), \(V\), and \(U\) are subspaces of \(\ell^{2}\), with \(V_N\) and \(U\) being closed and \(V\) not closed. The set \(W\) is not a well-defined subspace. The orthogonal spaces are \(V_N^\perp\), \(V^\perp\), and \(U^\perp\) respectively.

Step by step solution

01

Determine if \(V_N\) is a vector subspace

The vectors in \(V_N\) are those sequences which are zero after the N-th entry. We can verify that it contains the zero vector and is closed under vector addition and scalar multiplication, therefore \(V_N\) is a subspace of \(\ell^{2}\).
02

Determine if \(V_N\) is closed

A sequence in \(V_N\) converges to a limit in \(\ell^{2}\), and that limit is also in \(V_N\), therefore \(V_N\) is closed.
03

Find vectors orthogonal to \(V_N\)

The vectors orthogonal to \(V_N\) are those that have non-zero entries after the N-th entry. This set can be denoted as \(V_N^\perp\).
04

Determine if \(V\) is a vector subspace

The vectors in \(V\) are those sequences which are zero after some entry. Since \(V\) contains the zero vector and is closed under vector addition and scalar multiplication, \(V\) is a subspace of \(\ell^{2}\).
05

Determine if \(V\) is closed

A sequence in \(V\) may converge to a limit point where there are non-zero entries in places where they should be zero. This means the set \(V\) is not closed.
06

Find vectors orthogonal to \(V\)

The vectors orthogonal to \(V\) are those that have non-zero entries in a finite number of places. This set can be denoted as \(V^\perp\).
07

Determine if \(U\) is a vector subspace

The vectors in \(U\) are those sequences which are zero on even indices. Since \(U\) contains the zero vector and is closed under vector addition and scalar multiplication, \(U\) is a subspace of \(\ell^{2}\).
08

Determine if \(U\) is closed

A sequence in \(U\) converges to a limit in \(\ell^{2}\), and that limit is also in \(U\), therefore \(U\) is closed.
09

Find vectors orthogonal to \(U\)

The vectors orthogonal to \(U\) are those that have non-zero entries on even indices. This set can be denoted as \(U^\perp\).
10

Determine if \(W\) is a vector subspace

The vectors in \(W\) are those sequences which are zero at some index. The set of such sequences does not contain the zero vector unless the index is undefined, in which case the set is not well defined. Therefore \(W\) is not a subspace.
11

Find vectors orthogonal to \(W\)

Because \(W\) is not a well-defined set, it does not make sense to speak of vectors that are orthogonal to \(W\).

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Most popular questions from this chapter

If \(S\) is any subset of \(\mathcal{H}\), and \(V\) the closed subspace generated by \(S, V=\overline{L(S)}\), show that \(S^{1}=\left\\{u \in \mathcal{H} \mid\\{u|x\rangle=0\right.\) for all \(x \in S\\}=V^{1}\)

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 $$ \|\phi+\vartheta\|^{2}+\|\phi-\psi\|^{2}=2\|\phi\|^{2}+2\|\psi\|^{2} $$

For any pair of hermitian operators \(A\) and \(B\) on a Hilbert space \(\mathcal{H}\), define \(A \leq B\) iff \((u \mid A u) \leq(u \mid B u)\) for all \(u \in \mathcal{H}\). Show that this is a partial order on the set of hermatian operators pay particular attention to the symmetry property, \(A \leq B\) and \(B \leq A\) umplies \(A=B\). (a) For multiplication operators on \(L^{2}(X)\) show that \(A_{0} \leq A_{B}\). Iff \(\alpha(x) \leq \beta(x)\) a.e. on \(X\). (b) For projection operators show that the definition given here reduces to that given in the text, \(P_{N} \leq P_{N}\) iff \(M \subseteq N\)

Let \(\ell_{0}\) be the subset of \(\ell^{2}\) consisting of sequences with only finutely many terms different from zero. Show that \(\ell_{0}\) is a vector subspace of \(\ell^{2}\), but that it is not closed. What is its closure \(\overline{\ell_{0}} ?\)

Show that a vector subspace is a closed subset of \(\mathcal{H}\) with respect to the norm topology iff the limit of every sequence of vectors in \(V\) belongs to \(V\).
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