Characterization Of Lower Semi-continuity

Introduction

In the realm of real analysis and functional analysis, the concept of lower semi-continuity plays a crucial role in understanding the behavior of functions on Banach spaces. A function is said to be lower semi-continuous at a point if the set of points where the function is less than or equal to the value at that point is open. In this article, we will delve into the characterization of lower semi-continuity and explore the conditions under which a function is lower semi-continuous.

Preliminaries

Before we proceed with the characterization of lower semi-continuity, let us recall some essential definitions and concepts.

• A Banach space is a complete normed vector space, i.e., a vector space equipped with a norm that satisfies the following properties:
  • Positive definiteness: $\|x\| \geq 0$ for all $x \in X$ and $\|x\| = 0$ if and only if $x = 0$.
  • Homogeneity: $\|ax\| = |a| \|x\|$ for all $a \in \mathbb{R}$ and $x \in X$.
  • Triangle inequality: $\|x + y\| \leq \|x\| + \|y\|$ for all $x, y \in X$.
• A function $f: X \rightarrow \mathbb{R} \cup \{\pm \infty\}$ is said to be lower semi-continuous at $x \in X$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $f(y) \geq f(x) - \epsilon$ for all $y \in X$ with $\|y - x\| < \delta$.

Characterization of Lower Semi-Continuity

We are given a finite-dimensional Banach space $X$, a point $x \in X$, and a function $f: X \rightarrow \mathbb{R} \cup \{\pm \infty\}$. Our goal is to prove that $f$ is lower semi-continuous at $x$ if and only if it is continuous at $x$.

Lower Semi-Continuity Implies Continuity

Let us assume that $f$ is lower semi-continuous at $x$. We need to show that $f$ is continuous at $x$. To this end, let $\epsilon > 0$ be given. Since $f$ is lower semi-continuous at $x$, there exists a $\delta > 0$ such that $f(y) \geq f(x) - \epsilon$ for all $y \in X$ with $\|y - x\| < \delta$. Now, let $y \in X$ be such that $\|y - x\| < \delta$. We have

$$f(y) \geq f(x) - \epsilon \geq f(x) - f(x) + \epsilon = \epsilon.$$

On the other hand, since $f$ is lower semi-continuous at $x$, there exists a $\delta_1 > 0$ such that $f(z) \geq f(x) - \epsilon$ for all $z \in X$ with $\|z - x\| < \delta_1$. Let $z \in X$ such that $\|z - x\| < \delta_1$. We have

$$f(z) \geq f(x) - \epsilon \geq f(x) - f(x) + \epsilon = \epsilon.$$

Therefore, we have

$$\|f(y) - f(z)\| = |f(y) - f(z)| \leq |f(y) - \epsilon| + |f(z) - \epsilon| \leq |f(y) - f(x)| + |f(z) - f(x)| < 2\epsilon.$$

This shows that $f$ is continuous at $x$.

Continuity Implies Lower Semi-Continuity

Conversely, let us assume that $f$ is continuous at $x$. We need to show that $f$ is lower semi-continuous at $x$. To this end, let $\epsilon > 0$ be given. Since $f$ is continuous at $x$, there exists a $\delta > 0$ such that $\|f(y) - f(x)\| < \epsilon$ for all $y \in X$ with $\|y - x\| < \delta$. Now, let $y \in X$ be such that $\|y - x\| < \delta$. We have

$$f(y) \geq f(x) - \|f(y) - f(x)\| > f(x) - \epsilon.$$

This shows that $f$ is lower semi-continuous at $x$.

Conclusion

In this article, we have characterized lower semi-continuity in terms of continuity. We have shown that a function $f: X \rightarrow \mathbb{R} \cup \{\pm \infty\}$ is lower semi-continuous at $x \in X$ if and only if it is continuous at $x$. This result has far-reaching implications in real analysis and functional analysis, and it has been widely used in various applications.

References

• [1] Rudin, W. (1973). Functional Analysis. McGraw-Hill.
• [2] Kolmogorov, A. N., & Fomin, S. V. (1975). Introductory Real Analysis. Dover Publications.
• [3] Lang, S. (1993). Real and Functional Analysis. Springer-Verlag.

Further Reading

For further reading on the topic of lower semi-continuity and its applications, we recommend the following resources:

• [1] "Lower Semi-Continuity of Functions" by J. M. Borwein and A. S. Lewis (Journal of Mathematical Analysis and Applications, 1991)
• [2] "Semi-Continuity and Lower Semi-Continuity" by A. S. Lewis (Mathematical Surveys and Monographs, 1996)
• [3] "Functional Analysis and Applications" by S. R. S. Varadhan (Springer-Verlag, 2006)

Introduction

In our previous article, we explored the characterization of lower semi-continuity in terms of continuity. We showed that a function $f: X \rightarrow \mathbb{R} \cup \{\pm \infty\}$ is lower semi-continuous at $x \in X$ if and only if it is continuous at $x$. In this article, we will answer some frequently asked questions (FAQs) related to the characterization of lower semi-continuity.

Q1: What is the difference between lower semi-continuity and continuity?

A: Lower semi-continuity and continuity are two related but distinct concepts in real analysis and functional analysis. A function $f: X \rightarrow \mathbb{R} \cup \{\pm \infty\}$ is said to be lower semi-continuous at $x \in X$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $f(y) \geq f(x) - \epsilon$ for all $y \in X$ with $\|y - x\| < \delta$. On the other hand, a function $f: X \rightarrow \mathbb{R} \cup \{\pm \infty\}$ is said to be continuous at $x \in X$ if for every $\epsilon > 0$, there exists a $\delta > 0$ such that $\|f(y) - f(x)\| < \epsilon$ for all $y \in X$ with $\|y - x\| < \delta$.

Q2: Why is lower semi-continuity important in real analysis and functional analysis?

A: Lower semi-continuity is an important concept in real analysis and functional analysis because it has far-reaching implications in various areas of mathematics, such as optimization theory, calculus of variations, and partial differential equations. For example, the concept of lower semi-continuity is used to study the existence and uniqueness of solutions to optimization problems, as well as the behavior of functions on Banach spaces.

Q3: Can you provide an example of a function that is lower semi-continuous but not continuous?

A: Yes, consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = |x|$. This function is lower semi-continuous at every point $x \in \mathbb{R}$, but it is not continuous at $x = 0$.

Q4: How does the characterization of lower semi-continuity relate to the concept of convexity?

A: The characterization of lower semi-continuity is closely related to the concept of convexity. A function $f: X \rightarrow \mathbb{R} \cup \{\pm \infty\}$ is said to be convex if for every $x, y \in X$ and every $\lambda \in [0, 1]$, we have $f(\lambda x + (1 - \lambda) y) \leq \lambda f(x) + (1 - \lambda) f(y)$. It can be shown that a function $f: X \rightarrow \mathbb{R} \cup \{\pm \infty\}$ is convex if and only if it is lower semi-continuous.

Q5: Can you provide a reference for further reading on the topic of lower semi-continuity?

A: Yes, for further reading on the topic of lower semi-continuity, we recommend the following resources:

• [1] "Lower Semi-Continuity of Functions" by J. M. Borwein and A. S. Lewis (Journal of Mathematical Analysis and Applications, 1991)
• [2] "Semi-Continuity and Lower Semi-Continuity" by A. S. Lewis (Mathematical Surveys and Monographs, 1996)
• [3] "Functional Analysis and Applications" by S. R. S. Varadhan (Springer-Verlag, 2006)

We hope that this Q&A article has provided a comprehensive introduction to the characterization of lower semi-continuity. We encourage readers to explore the topic further and to apply the results to their own research and applications.