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# Advanced Linear Algebra MATH 5651

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This 20 page Class Notes was uploaded by Ila Haag on Monday September 21, 2015. The Class Notes belongs to MATH 5651 at University of Virginia taught by Michael Hill in Fall. Since its upload, it has received 27 views. For similar materials see /class/209576/math-5651-university-of-virginia in Mathematics (M) at University of Virginia.

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Date Created: 09/21/15

Hilbert Spaces A Math 551 Lecture November 28 2007 Outline 0 Setup amp Basic Examples 9 Convergence amp Cauchy Sequences a Hilbert Spaces What Do We Want 0 Want a nicer notion of basis for the infinite dimensional case 0 Should fit well with finite dimensional subspaces 0 Should be useable for computation The Key Example 2 0 Recall 2 X1iZiXIi2 lt 00 ltx7ygt G If x e 62 then limH00 Xn O O ltxn X1Xn0 then llxixnll a O 0 So with some idea of convergence any x is a limit of things we understand 0 Any x is a limit of a sequence of elements in e1 gt A Warning We must have a notion of convergence to do this A nonexample 0 Let V HHSNR 0 Inside V is V neNR 0 x 6 V thenwecan etxn X1XO as before BUT O we have no way to say that xn converges to x Moral We have to have convergence Metric a Convergence requires a notion of closeness o This is summed up in a topologywhich comes from a metric Detrir ition A metric space is a set X together with a function d Xx XHR suchthat 0 dXy 2 O with equality only ifX y 9 dX7y dy7X 9 dX7y S dX72 d27y Let V be an innerproduct space With dxy Hx 7 yHSaW in class that this satisfies the axioms for a metric This is our key example Let X be any set and let 1X7 y dX7y xy39 This is a metric space in Which every point is equally far from every other point so nothing is very close to anything else Segue Gotmvergeme A sequence in a metric space is defined just as for R an ordered list of elements We define convergence limits just as for R D t Let X1 be a sequence in a metric space This sequence converges to X if for every 5 gt 0 there is an N such that dXX lt e for all ngt N In other words X1 converges to X it the terms get closer and closer to X as n gets very large See at Two examples using the two metrics from before In an inner product space a sequence x converges to x if and only if Iim Hxni xii a o HANK Put another way we can see convergence in an inner product space by just checking convergence of the sequence Hxn 7 xi i E in With the other metric a sequence converges iff it is eventually constant In other words Xn converges to X if and only if there is some N such that for all n gt N Xn X We can see this by choosing e lt 1 Since all points are of uniform distance 1 the only points of distance less than this s are those equal to X iii In Rquot With the usual Euclidean distance a sequence of vectors x1 converges to x iff each coordinate converges This follows from the triangle inequality So inRz the sequence 3 1 3114 314141 31411414 converges to7r x This is the old rule from multivariable calculus JLngof1fknnlngo nnlngofk In 62 a sequence x1 converges if it converges coordinateWise This makes it easy to check here too Convergence has a big drawback you have to know the limit of a sequence This is extra data We want an intrinsic notion D r on A sequence Xn in a metric space is Cauchy if for all e gt 0 there is an N such that for all n m gt N dX Xm lt e Using the triangle inequality we can see that aquot convergent sequences are Cauchy A Warning In general NOTaII Cauchy sequences converge This is a property of the space LetX Q together With the metric dX y ix 7 yi The sequence 3 31 314 3141 is Cauchy It converges in R but it does not converge since 7139 Q Another example the sequence quot 1 XquotZW 1 is Cauchy It is not convergent In R it converges to a number that is actually transcendental Definition A metric space is complete if every Cauchy sequence converges o R and Rquot are complete 0 C and Cquot are complete 0 62 is complete 0 Q is not complete omplet ons The only incomplete metric space we have seen in Q and this sits inside a complete metric space R This is true in general 0 Every metric space embeds in a complete metric space 0 There is a smallest universal complete metric space that contains any given metric space This smallest complete metric space is called the completion If V is an inner product space then the completion of V is also an inner product space Definition A Hilbert space is an inner product space that is complete with respect to the induced metric So basically in a Hilbert space we can easily tell if sequences converge We have several key examples 0 Rquot and Cquot 0 2 Std 2 From convergence of sequences we get a notion of infinite series Letxn be a sequence and define SkX1 Xk If the sequence 5 converges to s then we say that the series Z x converges to s Just as for sums in R this is relatively weak and we might get a different answer by rearranging the sum Definition A series 2 xn is absolutely convergent if 2 Han converges complete and lute nee Absolute convergence is again something we check only in R In general absolute convergence only ensures that the sequence of partial sums is Cauchy We have the following theorem Let V be an inner product space Then V is complete if and only if absolute convergence implies convergence Since we will focus on Hilbert spaces we see that absolute convergence always implies convergence and we can again do all of our checks in R Hll E3 ll The following notions are equivalent for an orthogonal set 9 of vectors in a Hilbert space H 0 O is a Hilbert Basis o lfltvugt 0 forallu e O thenv 0 o Anyv is the limit of a sequence of vectors in the span of O The collection of all limits of all sequences of vectors in a set is called the closure of the set Since the span is a subspace the closure agrees with the completion of that subspace in the bigger complete subspace Jon In a Hilbert space this shows that a Hilbert basis is exactly what we want 0 we have convergence so we have infinite sums 0 any vector can be written as an infinite convergent linear combination of vectors of our basis We can do even betterwe can find the coefficients in this sum If 9 is a Hilbert Basis of a Hilbert space H and ifv e H then vv Zltvugtu ueO This is the Fourier expansion of v with respect to O

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