For any type of integer $n,$ the collection $left1, x, x^2, ldots, x^n ight$ is livirtually independent. Let $H_n$ be the expectancy of this collection, then $operatornamedim H_n=n+1 .$ given that $H_n$ is a subroom of $V,$ by theorem 11 on pg. $227, operatornamedim H_n=n leq operatornamedim V .$ given that we can construct $H_n$ for any kind of integer $n$ , the dimension of $V$ is better than every number. Therefore, the measurement of $V$ cannot be finite and also $V$ need to be infinite dimensional.
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Hello tbelow. Okay, So for this exercise, we should prove that the set off polynomial. Sorry. Hey, this is our I'm interpretation. This collection the set off polynomial actually corresponds. Finito, dimensional vector space. So exactly how to prove that? Okay, so it can be rather a interesting, yet how is identified the set of polynomial, So basically, we have the right to expand also. Okay, So instead of polynomial is basically the expectancy off one x x two additional square below X to the square X to the 3rd you desire till X m. Okay. So we deserve to specify, um, integral end. So in this instance, and coincides to an aspect off natural. Ah, on by the the the area off phenomenally identified by the span of these. So simply to recall that the spun indicates Elinor combinations off these facets right here on, it's clear that that corresponds to a paranormal off level and also appropriate, So, for instance, let's say that n equals 22 We acquired a X square plus b x plus c. So right here we obtained a direct combicountry off this aspects here until level to bereason we fix it and also equal to 2 okay on. In this situation, we're responding. The collection off point, but, were interested on spend the totality collection off polynomial and actually we confined Objection. So let's specify the following the following feature. So let's say we deserve to define a duty from the herbal to the set off polynomial B n. That is really straightforward surgeries human being to the P. That actually is really straightforward to construct this bereason we're going to take for any finish on the organic on. We're going to interpret into X to the finish. Okay. And we basically were constructing polynomial we discover, ah, attribute that take any type of any type of value off here from the organic here. So let's say 123 on we're map into the area off polynomial we're taking one. For example. On we obtain X to the one is an facet off. The point on this aacquire for 2 we deserve to map two x square for three is the very same X three okay on this F develop by ejection bereason for eincredibly number for eextremely element right here tright here exists one facet on the side of Paul Anonymous. So is one by one, correspondence on for ah, on eextremely eextremely eextremely facet. Here is map to the side of polygamy is incredibly subjective. So this attribute f is onto us objective. So that suggests that f responsive rejection So excellent. So we have characterized that f is a rejection on Actually, if you alert f develop a basis for the collection off for a Nahmias Why? Because we have actually a state prior to we have actually already watch that this expectancy at the collection, the space of pollen animals. So basically, we can say a basis for P will certainly be the basis for the space. P will be the set off features off finish. Such that end is an aspect off naturals. What I am placing into these methods basis is that fn Well, what is f one If one coincides to x f zero corresponds toe one and so on. So basically here in this shelp I'm putting one x x square to the third until extend But below we acquired somepoint And is any kind of facet off the natural aunt, we understand that the set off organic is an boundless set is a new infinite comfortable on we have fine Ah, by ejection f from n to the collection of polynomial. So that suggests on likewise f is rejected, so f is by objective. So that implies that and also at least, um, this off the set off Paula normals is, uh, has Dimension has actually the exact same measurement that end and also in this instance and actually objectives of have so have actually the same measurement on and also is an limitless collection. Soapy have to be on unlimited dimensional room as well, and that completes the proof.