Brief Summary
The video explores the concept of inner products using polynomials as examples. It begins by examining a potential inner product defined as the product of two polynomials evaluated at zero. The discussion covers the properties required for an inner product, such as commutativity, distributivity, and positive definiteness, and demonstrates how the initial example fails the positive definiteness test. The video also touches on how restricting the space of polynomials can affect whether a definition qualifies as an inner product, and draws a parallel between evaluating polynomials at infinitely many points and integration.
- A potential inner product defined as the product of two polynomials evaluated at zero is not positive definite.
- Restricting the space of polynomials can affect whether a definition qualifies as an inner product.
- Evaluating polynomials at infinitely many points is similar to integration, which brings positive definiteness back.
The video starts by considering whether the operation P(0) * Q(0) qualifies as an inner product for polynomials P and Q. This operation takes two polynomials, evaluates them at zero, and multiplies the results, producing a number. The properties of an inner product are then examined, starting with commutativity (or symmetry), which the operation trivially satisfies. Distributivity is also confirmed by considering the mechanics of evaluating Q as a sum of two functions.
The key property of positive definiteness is tested, and it's shown that P(0) * P(0) is not always positive. For example, if P(x) = x, then P(0) = 0, and P(0) * P(0) = 0, violating the positive definiteness requirement. Attempts to fix this by modifying the polynomial (e.g., x^2 - x) also fail, as other polynomials can be constructed to yield zero when evaluated using this operation.
The discussion introduces a subtlety: the same definition can be a legitimate inner product on a particular space but fail positive definiteness on a wider space. For instance, if the space is limited to quadratic polynomials, the previous counterexamples (which required cubic polynomials) are no longer valid. A quadratic polynomial cannot pass through three zeros, making the definition a valid inner product on this restricted space.
The video draws a parallel between evaluating polynomials at discrete points and integration. If infinitely many points are used (akin to an integral), positive definiteness can be restored, at least among reasonably continuous functions. An integral is like an infinite number of sums, and this brings positive definiteness back.
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Transcript
Full transcript of “"Almost Inner Products" on the Space of Polynomials”. Click a timestamp to open the video at that moment. Need another video? Use the YouTube transcript generator.
0:00 okay let's step back to pols and give you a couple of near examples near in a product examples for example P dotted with Q equals P of 0 time Q of 0 remember these are pols I can evaluate them if I want to right so instead of integ I can integrate them great that gives me a number I love it but I can also evaluate them who's to tell me that I cannot evaluate them and this is a rule that takes two pols produces a number so it's a candidate for inner product is it an inner product so I see
0:46 one hand up I want to see a couple more no you're saying you're shaking your head why not not positive definite yes uh symmetric clearly like when some or commutative rather so commutativity is the sort of thing that I don't even want to explain because when you start explaining it it invariably seems more complicated than it is so if you're really really confused about commutativity we'll sit down and really take take our time in going through what it means and it's more how you talk about it than what you actually do so I would describe this as of course it is symmetric of course it is commutative without explaining it just the word of
1:32 course I think explains it otherwise it becomes more complicated than it really is is it distributive of course it is distributive but don't just do it by staring at the equation and picking out the features of it go through in your head without writing things down the mechanics of what it would mean you have to explain it to yourself you have to say if I take Q to be a sum of two functions and I evaluated it in two different ways first dotted with the sum and otherwise dotted individually and then add together the results would I get the same thing and once you convince yourself that yes of course you would then you got it but positive definiteness Ben says no and so he has to give me a counter examples of two pols that I can multiply that where the
2:19 result will not be positive X not two pols one polom dotted with itself that would not be positive so what happens when I dot x with x what do I get zero cuz X evaluated at0 is Z yeah yeah zero okay so not positive definite so I'll fix it x^2 minus X where the hell did that come from x^2 - x yeah yeah he looked for something it was inspired by this and look for something that would have zero here and zero here so he said x * x -1 that's what it really
3:06 was according to this attempt at an inner product is zero okay I'll fix it did I fix it no because I can throw in time x - 2 oh I didn't say equals z * x - 2 correct so now I'll make one interesting point suppose that we limit ourselves to to the space of quadratic pooms isn't that interesting that on the space of quadratic pols this isn't in a product because you can't do this this cubic you can't have a quadratic polinomial go through three
3:52 zeros so that's another subtlety that one in the same definition can be a perfectly legitimate a product on a particular space and fail positive definit on a wider space so it's not too surprising but just make a mental note of it that it's quite possible for something like that to happen Okay so but if we're talking about polinomial in general there's no way I can fix it because you will always be able to find well by a clear recipe except do you kind of see that if I start throwing in points in the middle like like 05 and 75 and start filling in the line then it's kind of getting closer and closer to the integral right because that's what an integral would be
4:39 in this case between zero and two so I mean this is Idle talk but if I threw an infinitely many of these which of course there's no such thing because Infinity is a funny concept you know so I'm not saying that but I just want you to see the similarity between this and the integral definition that is an a product because an integral is sort of like in some ways in very specific narrow ways an infinite number of sums like these and that brings positive definiteness back at least among reasonably continuous functions
