91Ó°ÊÓ

In functional analysis, a normed space is a vector space on which a norm is defined. A norm is a function that assigns a non-negative length or size to each vector in the space. For example, if we have a vector space \(X\), a norm is usually denoted by \(\| \cdot \|\). Important properties of norms include:

- \( \| x \| \geq 0 \) (Non-negativity)
- \( \| x \| = 0 \) if and only if \( x = 0 \) (Definiteness)
- \( \| \alpha x \| = |\alpha| \cdot \| x \| \) for any scalar \( \alpha \) (Homogeneity)
- \( \| x + y \| \leq \| x \| + \| y \| \) (Triangle Inequality)

Normed spaces form a fundamental framework in which the concepts of distance and convergence can be defined. For instance, the unit sphere \( S_X \) in a normed space \( X \) is the set of all vectors with a norm of 1: \( S_X = \{ x \in X : \| x \| = 1 \} \). This concept is central to the given exercise because we work with sequences \( x_n \in S_X \) and \( y_n \in S_X \) leading to conclusions about their behavior.

A convex combination involves taking a weighted average of points where the weights sum up to 1 and are non-negative. For two points \( x \) and \( y \) on a line segment, a convex combination is given by \( z = \lambda x + (1 - \lambda) y \) where \( \lambda \in [0, 1] \). This ensures that \( z \) lies on the line segment connecting \( x \) and \( y \).

In the context of the exercise, \( z_n \) lies on the line segment between \( x_n \) and \( y_n \), which means \( z_n \) can be expressed as:
\( z_n = \lambda x_n + (1 - \lambda) y_n \),
where \( \lambda \in [0, 1] \). This representation is useful because it utilizes linearity and the properties of convex sets. For instance, the norm of a convex combination of unit vectors (our \( x_n \) and \( y_n \) are from the unit sphere) does not exceed 1. This aligns with the step-by-step logic provided: using the triangle inequality to justify that \( \| z_n \| \leq 1 \) and leveraging the specific properties of norms.

Limits play a vital role in functional analysis, especially in the context of sequences in normed spaces. A sequence \( \{ x_n \} \) is said to converge to a limit \( x \) if for every \( \epsilon > 0 \), there exists a natural number \( N \) such that \( \| x_n - x \| < \epsilon \) for all \( n > N \). This means that the distance between \( x_n \) and \( x \) becomes arbitrarily small as \( n \) increases.

For the given problem, we know:
- \( \lim_{n \to \infty} \| x_n + y_n \| = 2 \).
- \( \{ x_n \} \) and \( \{ y_n \} \) are sequences in the unit sphere, implying \( \| x_n \| = 1 \) and \( \| y_n \| = 1 \).

This information leads us to understand the behavior of \( \{ z_n \} \):

- Since \( \| x_n + y_n \| = 2 \), \( x_n \) and \( y_n \) are very close to being collinear, meaning they almost point in the exact same direction.
- This close alignment and the convex combination property squeeze \( z_n \) to maintain its norm close to 1. Eventually, as \( n \to \infty \), \( z_n \rightarrow 1 \).

Thus, by understanding these limit properties, we can conclude the required result: \( \lim_{n \to \infty} \| z_n \| = 1 \).