Measure Theory/Integral Product Bound and Triangle Inequality
The Lp Product Bound
[edit | edit source]Now recall that the point of obtaining the real product-to-sum bound from the previous lesson,
- for any and and
was in the hope of obtaining a bound on an integral of the form .
Exercise 1. Integral Product Bound
Let and and then let and . Prove that . Hint: In many cases of a normed vector space, it is easier to prove a claim first for vectors of unit norm, and then relate this result to the space of all vectors. In this case, that approach can help to simplify the proof. So start by assuming that , in which case your goal is to prove
Now apply the sum-to-product bound. When you have finished that, you now need to let have any norm at all. Consider first the case that either f or g has norm equal to zero, which should be straight-forward. Next consider that their norms are both not zero and consider the normalized vectors . Argue that these two vectors now each have norm equal to 1 and then apply your result from before. Then with a little bit of manipulation, infer the desired result. |
Exercise 2. The Conjugate Function
come out correct. The function which does so, is sometimes called the conjugate of f. 1. Prove that .
2. Prove that .
3. Show that .
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The Triangle Inequality
[edit | edit source]Finally we are in a position to prove the triangle inequality for .
Exercise 3. The Lp Triangle Inequlity
For prove that . Hint: From split off a single factor of and then use this to distribute the multiplication and then the integration. What is left should be a conjugate function for . Continue from there. |
Exercise 4. Lp Is Complete
Show that the proof for completeness is exactly the same proof for completeness, mutatis mutandis. |