Question

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

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

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 \(\ell^{2}\).

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.

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 \(V_N^\perp\).

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 \(\ell^{2}\).

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.

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 \(V^\perp\).

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 \(\ell^{2}\).

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.

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 \(U^\perp\).

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.

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 \(W\).

Key concepts

Vector Subspaces

In the realm of functional analysis, a vector subspace is a subset of a vector space that satisfies two main properties: it must contain the zero vector and must be closed under vector addition and scalar multiplication. This means if you take any two vectors in the subspace and add them together, or if you multiply any vector by a scalar, the result should also be within the same subspace.

In our exercise, some subspaces like \(V_N\) and \(U\) meet these criteria because their vectors follow specific rules regarding their components. For instance, \(V_N\) includes sequences which become zero after a certain point, while \(U\) consists of sequences that are zero at even indices. Each satisfies the key conditions of vector subspaces. Understanding these properties is essential for identifying subspaces when working with infinite-dimensional vector spaces like \(\ell^2\).

Orthogonality

Orthogonality is a concept that depicts a condition of being perpendicular. In vector spaces, two vectors are orthogonal if their dot product equals zero. When dealing with sequences in spaces like \(\ell^2\), vectors orthogonal to each other have no overlap in directions of their respective components that contribute to their magnitude when viewed through inner products.

Consider \(V_N\) from our exercise. Vectors orthogonal to \(V_N\) are those that have non-zero components where \(V_N\)'s vectors are zero—particularly after the \(N\)-th position. This concept of orthogonality plays a pivotal role when resolving components of vectors in different directions, ensuring that these components do not interfere with each other. It allows for an elegant partitioning of vector spaces.

Hilbert Space

A Hilbert space is a special kind of vector space that is complete with respect to the distance function defined by its inner product. Completeness means that every Cauchy sequence of vectors in the space has a limit that is also within the space.

In our exercise base, \(\ell^2\) serves as a quintessential example of a Hilbert space. This is a space of square-summable sequences, meaning that if you take the squared magnitudes of the sequence members and sum them up, the sum remains finite. The structure of Hilbert spaces equips us with tools for discussing convergence, orthogonality, and the projection of vectors onto subspaces, which are essential when analyzing infinite-dimensional spaces.

Because of these properties, Hilbert spaces are crucial tools in areas such as quantum mechanics and signal processing, where concepts of distance and orthogonality lead to practical applications and solutions.

Closed Sets

In the context of vector spaces, closed sets have a particular meaning: a set is closed if it contains all its limit points, meaning any sequence of vectors in the set that converges will have its limit also within the set. This property is vital for many mathematical analyses, ensuring that the behavior of sequences is predictable and that operations such as limits can be taken without leaving the confines of the original set.

For example, \(V_N\) and \(U\) in the exercise are indeed closed sets because sequences that conform to their defining properties won't "escape" or reach a limit that falls outside the set. Conversely, \(V\) is not closed because it's constructed in a way that allows sequences to reach limit points with entries that should be zero but aren't. Knowing whether a set is closed can determine the kind of mathematical operations and theorems that can be applied to that set, which is essential in rigorous mathematical proofs and real-world applications.