Question

Let \(X\) be a Banach space. Show that if \(E, F\) are finite-dimensional subspaces of \(X\) such that \(\operatorname{dim}(F)>\operatorname{dim}(E)\) (no assumption on inclusion of \(E\) and \(F)\), then there is \(x \in F\) with \(\|x\|=\operatorname{dist}(x, E)=1\).

Short answer

There exists an element \(x \in F\) such that \(\|x\| = \text{dist}(x, E) = 1\).

Step-by-step solution

Understand the Problem

We need to show that for finite-dimensional subspaces E and F of a Banach space X, where the dimension of F is strictly greater than that of E, there exists an element x in F such that x has norm 1 and its distance to E is also 1.

Define Distance from a Point to a Subspace

Recall that for a point x in a Banach space X and a subspace E of X, the distance from x to E is defined as \(\text{dist}(x, E) = \text{inf} \{ \|x - e\| : e \in E \}\).

Consider the Unit Sphere in F and E

Look at the unit sphere in F, which is defined as \(S_F = \{y \in F : \|y\| = 1\}\). Since F is finite-dimensional, S_F is compact.

Note the Separation of Subspaces

Since \(\text{dim}(F) > \text{dim}(E)\), and F and E are finite-dimensional subspaces, there must be some element of F that cannot be part of E due to their different dimensionalities.

Define Maximum Distance to E for Elements in F

Consider the function \(d(y) = \text{dist}(y, E)\) for \(|y| = 1\), which is continuous by the compactness argument, and is defined on the compact set \(S_F\).

Find the Maximum Value of the Distance Function

By the extreme value theorem, the continuous function \(d(y)\) attains a maximum on the compact set \(S_F\). Let \(x \in S_F\) be such that \(d(x)\) is maximum.

Verify the Norm and Maximum Distance

By construction, \(d(x) = \text{dist}(x, E)\) and \(\|x\| = 1\). Since \(d(x)\) is maximized, and because \(\|x\| = 1\), the maximum distance must be at least 1. Hence, there must be an element \(x \in F\) such that \(\|x\| = \text{dist}(x, E) = 1\).

Key concepts

Finite-Dimensional Subspaces

Finite-dimensional subspaces are simply subspaces of a Banach space with a finite basis. This implies that every element in these subspaces can be represented as a linear combination of a finite number of vectors. Such subspaces are important in various contexts because they exhibit properties that general infinite-dimensional spaces do not. In our context, we consider subspaces E and F of a Banach space X, where \(\text{dim}(F) > \text{dim}(E)\). Due to the dimensional difference, there's more 'room' in F, which implies there exist elements in F not in E.

Distance in Banach Space

The distance between a point and a subspace in a Banach space is defined using an infimum. For a point x and subspace E, it is given by \(\text{dist}(x, E) = \text{inf} \{ \|x - e\| : e \in E \}\). This essentially measures how 'close' the point is to the subspace. By identifying the minimum norm of the difference between x and any element e from E, we ascertain the closest approach of x to E. This concept proves essential when evaluating the problem's condition that \(\text{\text{dim}(F) > \text{\text{dim}(E)}}\).

Extreme Value Theorem

The Extreme Value Theorem is pivotal in our proof. It states that if a function is continuous on a compact set, it attains a maximum and a minimum within that set. In our case, consider the function d(y) = dist(y, E), defined on S_F, the unit sphere in F, which is compact because F is finite-dimensional. This theorem guarantees that d(y) achieves its supreme value at some point y in S_F. Hence, by the maximum value, we locate such an x in F where both \(\text{dist}(x, E) = 1\) and \(\text{\|x\| = 1\).

Unit Sphere

A unit sphere in a Banach space (or specifically in subspace F) is the set of all points (elements) that have a norm of 1. Mathematically, it is defined as \['S_F = \{y \in F : \|y\| = 1\}\]. The unit sphere is important in our proof because it provides a compact set where we apply the Extreme Value Theorem. The compactness is a direct result of F being finite-dimensional, ensuring the existence of extrema for continuous functions defined on it.

Functional Analysis

Functional analysis is the broader field of mathematics we are working within. It deals with spaces of functions and the study of functional spaces, including concepts like Banach spaces, Hilbert spaces, operators, and norms. In the context of this problem, we leverage several essential results from functional analysis such as properties of finite-dimensional spaces, concepts of norm and distance, as well as the Extreme Value Theorem. Understanding these foundational ideas makes it easier to grasp this specific problem as well as other more complicated scenarios in functional analysis.