Prove that if [a, b] is a bounded interval and f ∈C^0[a, b], then f2 ≤√(b-a)f∞.

Name: Problem 39, Chapter 3: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2021-12-21T09:57:38-08:00
Duration: 4 min 10 s
Channel: Harmender Singh Yadav
Description: Prove that if [a, b] is a bounded interval and f ∈C^0[a, b], then f2 ≤√(b-a)f∞.

Prove that if [a, b] is a bounded interval and f ∈C^0[a, b],...

Key Concepts

Bounded Interval

A bounded interval is a segment of the real line with finite length. In analysis, ensuring the interval is bounded implies that measures like length and integration limits are finite, which is crucial when considering norms defined via integrals over the interval.

Continuous Function on a Closed Interval

A function that is continuous on a closed interval [a, b] is guaranteed to achieve its maximum and minimum values due to the extreme value theorem. This property is essential because it allows us to define the supremum norm (or infinity norm) and ensures that the function is bounded, thereby making the integral norms well-defined.

L2 Norm

The L2 norm, also known as the Euclidean norm in function spaces, is defined as the square root of the integral of the square of the function over the interval. It provides a measure of the function's magnitude in space while accounting for the distribution of the function's values across the domain.

Supremum Norm

The supremum norm, or L? norm, is defined as the maximum absolute value that the function attains on the interval. It gives a uniform bound for the function and is particularly useful in providing bounds for other norms when the domain is of finite measure.

Relationship between L2 Norm and Supremum Norm on Finite Measure Spaces

In finite measure spaces, such as a bounded interval, there exists an inequality that relates different Lp norms. Specifically, the L2 norm can be bounded by the supremum norm multiplied by the square root of the measure of the domain. This inequality is an application of the property that the integral of a non-negative function over a finite set is at most the measure of the set times the maximum value of the function.

Transcript

00:01 In this question, it is given that the function f is continuous on the close interval a to b, and f of x is always greater than or equal to 0 on the interval, that is x belonging to the interval a to b.

00:15 Now, we need to prove that the integral of the function f of x for the limit a to b is also greater than or equal to 0.

00:22 So, from the definition of the definite integral, we know that that is the integral of function f of x dot d x for the limit a to b is equal to actually that is the limit we can write it as the that is sum of limit limit of the sum that is limit is n tends to infinite for that is summation of f of u i dot delta x that is d x or delta x here and where i is from 1 to n so this is the from the definition of definite integrals we can write the integral of f of x definite integral of x for the limit a to b as the limit of sum sum of f of ui times delta x where i is from 1 to n and n tends to infinite this is the limit now we know as the function f of x is greater than 0 and also that is delta x is also greater than 0 so we will have as f of x greater than r equal to 0 and delta x will also be greater than 0 so we will have that the summation of this f of ui delta x will also greater than 0 so from here we get that that is summation for f of ui times delta x for i 1 to n will also be greater than or equal to 0 therefore by the that is properties of the limits we will have this that is the summation of that is limit of limit for n tends to infinite that is summation of f of ui dot delta x for i equals to to 1 to n this will be greater than or equal to limit n tends to infinite on 0, the limit on 0 which is also equal to 0 here.

02:53 So on the left hand side we know that this definition actually corresponds to the integral of that is f of x .d .x we can apply the limits as a to b...

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