Question

Let \(R\) be a normed linear space. Prove that a) Every finite-dimensional linear subspace of \(R\) is closed; b) If \(M\) is a closed subspace of \(R\) and \(N\) a finite-dimensional subspace of \(R\), then the set $$ M+N=\\{z: z=x+y, x \in M, y \in N\\} $$ is a closed subspace of \(R\); c) If \(\mathrm{Q}\) is an open convex set in \(\mathrm{R}\) and \(x_{0} \notin \mathrm{Q}\), then there exists a closed hyperplane which passes through the point \(x_{0}\) and does not intersect \(Q .\)

Short answer

Finite-dimensional subspaces of normed spaces are closed; sums of closed and finite-dimensional subspaces are closed; a hyperplane can exist to separate a point from an open convex set.

Step-by-step solution

Understanding Subspaces in Normed Linear Space

A normed linear space is a vector space equipped with a function called a norm, which assigns a strictly positive length or size to each vector in the space. A linear subspace is a subset of this space that itself forms a vector space under the same operations. We want to show that a finite-dimensional linear subspace of a normed linear space is closed. This means that every Cauchy sequence of points in this subspace will converge to a point that is also within this subspace.

Proving (a) - Finite-dimensional subspaces are closed

Let \( V \) be a finite-dimensional subspace of a normed linear space \( R \). Given a Cauchy sequence \( \{x_n\} \) in \( V \), it must converge to a limit \( x \in R \) because \( R \) is a normed space. Since \( V \) is finite-dimensional, it is complete and hence \( x \) is in \( V \). Therefore, \( V \) is closed in \( R \).

Proving (b) - Sum of closed subspace and finite-dimensional subspace is closed

Consider closed subspace \( M \) and finite-dimensional subspace \( N \). Let \( \{z_n\} \) be a converging sequence in \( M+N \), such that \( z_n = x_n + y_n \), where \( x_n \in M \) and \( y_n \in N \). \( \{y_n\} \) being inside finite-dimensional \( N \) must also be a converging sequence to some \( y \in N \). Due to closing properties of \( M \), \( \{x_n = z_n - y_n\} \) converges in \( M \) to some \( x \). Hence, the limit \( z = x + y \) remains in \( M+N \), confirming \( M+N \) is closed.

Understanding convex sets and hyperplanes

A convex set is a set where any line segment connecting two points in the set lies entirely within the set. An open convex set does not include its boundary. A hyperplane in a normed space \( R \) is a flat subset whose dimension is one less than that of \( R \). We need to find a hyperplane through \( x_0 \) that doesn’t touch the convex set \( Q \).

Proving (c) - Existence of a separating hyperplane

Since \( x_0 otin Q \), we can apply the supporting hyperplane theorem, which states that there exists a hyperplane that separates \( x_0 \) from \( Q \). This hyperplane can be represented as \( \langle a, x \rangle = \langle a, x_0 \rangle \) for some normed vector \( a \in R \). Because \( Q \) is convex and open, and \( x_0 \) is outside, \( Q \) will completely lie on one side of this hyperplane, hence the hyperplane is closed and does not intersect \( Q \).

Key concepts

Finite-dimensional Subspaces

In normed linear spaces, subspaces are subsets that themselves follow the rules of a vector space. Finite-dimensional subspaces are special because they have a limited number of directions or basis vectors. When we say a subspace is closed, we're essentially saying any Cauchy sequence of vectors within that subspace will converge to a vector that also belongs to the subspace.

Why does this matter? Because finite-dimensional spaces are complete by nature. They are like comforting self-contained ecosystems where everything stays neatly within. This means if your journey started within the subspace, you will end within it too, even at the limit. That's why every finite-dimensional subspace is closed in a normed linear space.

Closed Subspaces

Closed subspaces in a normed linear space are subspaces that contain all their limit points. If you have a sequence of vectors that converges, the limit lies conveniently within the same set. This is an essential property as it allows safe mathematical operations knowing limits won't escape the subspace.

Now, what if we have two subspaces, one being closed and the other finite-dimensional? Their sum, which combines elements from both, remains closed. Why? Because any converging sequence from this sum can be broken down into parts converging individually within the original spaces, respecting their closed property and ensuring the overall sequence will not reach out.

• This results in the sum of the closed and finite-dimensional subspaces staying inside our original normed space without losing convergence.

Convex Sets

Convex sets in a normed space are intuitively simple: pick any two points inside them, and the line connecting them sits entirely within the set. Imagine such a set as a stretchy, flexible sheet that can bend but won't break the connectivity.

Open convex sets, those that don't include edge boundaries, are particularly interesting. They offer a playground where separating techniques can be applied, as we'll see when discussing hyperplanes. These sets are essential in optimization problems, as they ensure there's a direct, unobstructed path to the optimum without getting stuck in local minima.

Supporting Hyperplane Theorem

The supporting hyperplane theorem is an elegant and practical mathematical tool. In essence, it allows us to "draw" a flat, sheet-like boundary that can help separate different sets in a normed linear space.

How does it work? Imagine you have a convex set, and a point not inside it. The theorem promises that there exists a hyperplane that neatly places this point on one side and the entire convex set safely on the other. This hyperplane doesn't intersect the set, acting like a perfect separator.

• This ability to separate adds great utility, especially in optimization and decision-making problems where constraints need to be clearly defined, without commingling different regions.

By placing barriers methodically, it provides clarity and precision in complex spaces.