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Chapter 10: Problem 15

If \(f, g \in \mathcal{L}[a, b]\), show that \(|\|f\|-\|g\|| \leq\|f \pm g\|\).

Short Answer

Expert verified
The property \(| \|f\| - \|g\| | \leq \|f \pm g\|\) of norms in the space of integrable functions can be proven by making an analogy between this property and the absolute value inequalities valid in real numbers, and then breaking down the right side of inequality into its integral expression. This results in the complete proof of the given inequality.

Step by step solution

01

Present the inequality

From the properties of absolute value, it is known that for any real numbers \(x\) and \(y\), \(|x - y| \leq |x + y|\). We can make the analogy where \(x = \|f\|\) and \(y = \|g\|\). Then we get \(| \|f\| - \|g\| | \leq \|f\| + \|g\|\).
02

Break down the right side of inequality

Thinking about what the right side of the inequality means in terms of the original integral expressions. \( \|f\| + \|g\| = \int_a^b |f(x)| dx + \int_a^b |g(x)| dx = \int_a^b (|f(x)| + |g(x)|) dx\). Since \(|f(x)|\) and \(|g(x)|\) are both nonnegative, their sum is still a nonnegative function that is integrable over the interval \( [a, b]\). Thus, by properties of absolute value, \(\int_a^b (|f(x)| + |g(x)|) dx \geq \int_a^b |f(x) + g(x)| dx = \|f + g\|\).
03

Completion of proof

By substituting this result back into the inequality derived in Step 1, we have \(| \|f\| - \|g\| | \leq \|f + g\|\). Similarly, using the same logic on \(| \|f\| - \|g\| | \leq \|f\| + \|g\|\), we can replace \(f\) with \(-g\) to get \(| \|f\| - \|g\| | \leq \|f - g\|\). Therefore, we have proven that \(| \|f\| - \|g\| | \leq \|f \pm g\|\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value Inequality
The absolute value inequality frequently appears in real analysis and is a fundamental tool in proving various mathematical statements. For any real numbers \( x \) and \( y \), the inequality \( |x - y| \leq |x + y| \) holds. This tells us that the absolute difference between two values cannot exceed the sum of their absolute values.

In the context of functions, particularly integrable functions on an interval \([a, b]\), this concept is applied to norms of functions, symbolized as \( \|f\| \) for a function \( f(x) \). The norm \( \|f\| \) is typically defined as the integral over the absolute value of the function, \( \|f\| = \int_a^b |f(x)| \, dx \).

With this understanding, replacing \( x \) and \( y \) with \( \|f\| \) and \( \|g\| \) respectively, gives us \( |\|f\| - \|g\|| \leq |\|f\| + \|g\|| \), illustrating that the difference of the norms cannot exceed the norm of the sum or difference of functions.
Integral Expressions
Integral expressions can seem complex, but they simply represent the total "accumulated value" of a function over an interval. In the case of real-valued functions, this involves summing the area between the curve and the x-axis over a defined range, usually \([a, b]\).

For an integrable function \( f \), the expression \( \int_a^b |f(x)| \, dx \) denotes the integral of the absolute value of \( f \) over the interval \([a, b]\). This equation represents the "norm" or "magnitude" of the function's integral, disregarding whether the function dips below the x-axis.

When considering the sum of two integrable functions \( f \) and \( g \), the combined integral \( \int_a^b (|f(x)| + |g(x)|) \, dx \) still defines a valid integral. Importantly, from our initial problem we see: \( |\|f\| - \|g\|| \leq \int_a^b (|f(x) + g(x)|) \, dx = \|f + g\| \), reaffirming that integral expressions must be managed considering absolute values, especially to maintain inequality relations.
Properties of Integrable Functions
The properties of integrable functions are key when discussing their behavior over an interval. If a function is integrable over \([a, b]\), it means that you can take the integral of that function over the interval, and the result will be a finite value.

A vital characteristic is their ability to maintain additive properties under integration. This means if \( f \) and \( g \) are integrable, then both \( f + g \) and \( f - g \) are also integrable. This characteristic is pivotal in proving multiple function inequalities, especially involving absolute values.

Moreover, if a function \( f(x) \) is continuous over the interval \([a, b]\), it is guaranteed to be integrable on that range. Continuity is a significant friend to integrability, providing an assurance of no "jumps" or "gaps" in the function, making its integral finite and well-defined.
  • Additivity: The integral of a sum is the sum of the integrals, \( \int_a^b (f(x) + g(x)) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx \).
  • Non-negativity of absolute integrals: \( \int_a^b |f(x)| \, dx \) is non-negative because it represents a "distance" of sorts.
These properties ensure the methods we use in real analysis lead to precise and logical conclusions.

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Most popular questions from this chapter

(a) Give an example of a function \(f \in \mathcal{R}^{*}[0,1]\) such that max \(\\{f, 0\\}\) does not belong to \(\mathcal{R}^{*}[0,1]\). (b) Can you give an example of \(f \in \mathcal{L}[0,1]\) such that \(\max \\{f, 0\\} \notin \mathcal{L}[0,1] ?\)

With the notation as in Exercise 17, the Chartier-Dirichlet Test asserts that if \(f \in \mathcal{R}^{*}[a, \gamma]\) for all \(\gamma \geq a\), if \(F(x):=\int_{a}^{x} f\) is bounded on \([a, \infty)\), and if \(\varphi\) is monotone and \(\lim _{x \rightarrow \infty} \varphi(x)=0\), then \(f \varphi \in \mathcal{R}^{*}[a, \infty] .\) (a) Show that the integral \(\int_{0}^{\infty}(1 / x) \sin x d x\) converges. (b) Show that \(\int_{2}^{\infty}(1 / \ln x) \sin x d x\) converges. (c) Show that \(\int_{0}^{\infty}(1 / \sqrt{x}) \cos x d x\) converges. (d) Show that the Chartier-Dirichlet Test does not apply to establish the convergence of \(\int_{0}^{\infty}(x /(x+1)) \sin \left(x^{2}\right) d x\) by taking \(f(x):=\sin \left(x^{2}\right)\)

Let \(h_{n}(x):=n\) for \(x \in(0,1 / n)\) and \(h_{n}(x):=0\) elsewhere in \([0,1]\). Does there exist \(h \in \mathcal{L}[0,1]\) such that \(\left\|h_{n}-h\right\| \rightarrow 0 ?\)

Establish the convergence or the divergence of the following integrals: (a) \(\int_{0}^{\infty} \frac{\ln x d x}{x^{2}+1}\), (b) \(\int_{0}^{\infty} \frac{\ln x d x}{\sqrt{x^{2}+1}}\), (c) \(\int_{0}^{\infty} \frac{d x}{x(x+1)}\), (d) \(\int_{0}^{\infty} \frac{x d x}{(x+1)^{3}}\), (e) \(\int_{0}^{\infty} \frac{d x}{\sqrt[3]{1+x^{3}}}\) (f) \(\int_{0}^{\infty} \frac{\operatorname{Arctan} x d x}{x^{3 / 2}+1}\).

Suppose \(I \subseteq \mathbb{R}\) is a closed interval and that \(f:[a, b] \times I \rightarrow \mathbb{R}\) is such that \(\partial f / \partial t\) exists on \([a, b] \times\) \(I\), and for each \(t \in[a, b]\) the function \(x \mapsto f(t, x)\) is in \(\mathcal{R}^{*}(I)\) and there exist \(\alpha, \omega \in \mathcal{R}^{*}(I)\) such that the partial derivative satisfies \(\alpha(x) \leq \partial f(t, x) / \partial t \leq \omega(x)\) for a.e. \(x \in I .\) If \(F(t):=\int_{I} f(t, x) d x\), show that \(F\) is differentiable on \([a, b]\) and that \(F^{\prime}(t)=\int_{l} \partial f(t, x) / \partial t d x\).
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