Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such that: T c, D { T } Y X T : D T Y be a liear operator The costat c ay be viewed as the upper boud of the aplificatio of or produced by the operator T (I the sequel we drop the subscripts i i ad i ) Clearly, c,, ad the sallest c such that this iequality holds DT { }, is T equal to the supreu of This quatity is the iduced operator or, or just operator or: X Y T : = sup D{ T } θ Note that with c = T, we have T Lea: Let T be a bouded liear operator as defied above The: (a) T = sup D{ T } = (b) The quatity T is a or Proof: (a) Write = α ad set z = The α z = = α T : = sup = sup = sup T = sup z α α T { } { } { } z { } θ θ θ z = L- /8
(b) Check that T satisfies the properties of a or Eaples: (a) Itegral Operator ( Cab ) ( Cab ) T : [, ], i [, ], i, y = T where C C b yt () = ( τ) ht (, τ) dτ a Here h(,) is called the kerel (or ipulse respose) of T ad is assued to be cotiuous o the closed square G : = [ ab, ] [ ab, ] This type of syste is liear tie-varyig t () yt () htτ (, ) It reais to be show that htτ (, ) is bouded Sice h(,) is cotiuous o the closed square, it is bouded, by say, ht (, τ) k, (, tτ) G, k R We have: t () a t () = c t [ a, b] Thus, b y = = a ( τ) h( t, τ) dτ a ( τ) h( t, τ) dτ k ( b a) c c t [ a, b] t [ a, b] c a a b Therefore the itegral operator is bouded by: (b) Fiite-Diesioal Trasforatios T k ( b a ), C [ a, b ] c c L- /8
a a The real atri A = defies a fiite-diesioal liear operator T : E a a with y = A The copoets of the output vector ca be writte as: E y = α, j =,, j jk k k = We kow that T is liear ad we ow show that it is bouded Recall that the or o the Euclidea / vector space E is = i, ad siilarly for y E Thus, i= = yj = α jkk j= j= k= j j= j= α j α jk j= j= k= = α, α (Cauchy-Schwarz Ieq) = = j Hece the liear fiite-diesioal atri operator is bouded:,, = jk j= k= c E c α It is a fudaetal ad iportat fact that for liear operators, cotiuity ad boudedess becoe equivalet properties Theore: Let T : D{ T } Y be liear, where D{ T } X, ad XY, are ored liear spaces The: (a) T is cotiuous it is bouded (b) If T is cotiuous at a sigle poit, it is cotiuous Proof: (a) For T =, the stateet is trivial Let T T ( ) Assue T is bouded ad cosider ay DT ( ) Let ε> be give Sice T is liear, ε for every DT ( ) such that < δ, where < δ =, we obtai: T L- 3/8
= T ( ) T < T δ = ε Sice DT ( ) was arbitrary, we coclude that T is cotiuous ( ) Assue T is cotiuous at a arbitrary DT ( ) The, give ε>, there is a δ> such that T ε, DT ( ) satisfyig δ Take ay z DT ( ) ad set: δ δ = + z = z z z = δ The: Hece T is bouded δ δ ε = T ( ) T ( z) = Tz z z ε z Tz δ (b) Cotiuity of T at oe poit iplies boudedess (by the ( ) part of the proof of (a)), which i turs iplies cotiuity of T by (a) 35 Hilbert Spaces Revisited We eed to itroduce additioal properties of Hilbert spaces, which we will use i our discussio of cotrollability operators Propositio: The ier product ii, : V V F of a ier product space is a cotiuous fuctio Equivaletly, if ad y y, the, y, y that the or is also a cotiuous fuctio I particular, iplies, so L- 4/8
Proof: F R First, if { } is a coverget sequece, the { } Let = is bouded, eg, K < (choose N such that <, N ad let K = + + a i ) Now, i N y, = ( ) +,( y y) + y =, y y +, y +, y y +, y The first three ters o the RHS coverge to zero, eg,, y y y y K y y < K for soe Hece,, y, y For cotiuity, we eed to show that for ay y, V, ad give ε>, there eists δ for all, y satisfyig y y = + y y < δ, y, y < ε This ca be prove by creatig sequeces { }, { } y y takig o the values, soe k ad such that, y, y < ε Such sequeces ca always be foud k k y for The ost iportat cocept i Hilbert space is orthogoality I a Hilbert space, whe we cosider subspaces, we ake sure they are closed subspaces, ie, they cotai all liits of sequeces i the 35 Subspace of a Hilbert Space Let H deote a Hilbert space The M (i) y, M, αβ, F α+ βy M, (ii) { } Notes: is a Cauchy sequece i M = li M H is called a subspace of H if the followig hold: + (a) Note that the oly additioal coditio fro previous defiitios of subspaces is (ii) (b) Subspaces of R are autoatically closed, but this is ot true for every Hilbert space L- 5/8
35 Orthogoality i Hilbert Space Defiitio: Two vectors y, H are orthogoal, deoted by y, if y, = If Y H is ay subset, the Y eas y, y Y We shall deote by LY ( ) the sallest closed subspace cotaiig Y Cotiuity of the ier product ii, yields Y L( Y ) Defiitio: The orthogoal copleet of ay Y H is the set of vectors that are orthogoal to all vectors i Y : = H: y, y Y Y,ie, { } Note that Y is a subspace of H The direct su of two orthogoal subspaces XY, H is defied by X Y If Y = X, the H = X Y= X X Theore: Let H be a Hilbert space, ad M H be a subspace The, ay vector H has the uique decopositio = y+ z, where y M, z M Furtherore, y = i v v M Proof: First, cosider the uiqueess of the decopositio Suppose two pairs of vectors yz, H ad y, z H have the asserted property The: = y+ z = y + z y y = z z, y y M, z z M y y, y y = z z, z z = y y, z z = y y =, z z = Hece, y = y, z = z Now if M, the its decopositio is trivial: = + θ, so we assue that M, i which case: if v = i v = h >, v M v M L- 6/8
sice M is closed Let { } parallelogra idetity: 34-5 LINEAR SYSTEMS y be a sequece i M such that y h Recall the u+ v + u v = u + v, u, v H Let u = y, v = y, the y y + y y = y + y Now, sice y + y M, y y = 4 ( y + y ) 4h Fi ε> ad choose N such that for all, > N, The, fro the above parallelogra idetity, y < ε + h, y < ε + h 4 4 Thus, { } y y = y + y y y y y ( ε + h ) + ( ε + h ) 4h = ε 4 4 y is a Cauchy sequece i M, ad hece there eists a liit y M, y y Furtherore, y = h by cotiuity of the ier product It reais to show that y M Suppose ot for a cotradictio The, there eists w M y, w = r > Now, for ay c, y+ cw M so that: y+ cw h = y Thus, L- 7/8
y+ cw y = y c y, w + c w y c y w c w =, + r = y, w < c w But sice the costat c was arbitrary, this last iequality leads to a cotradictio whe we take r c = Therefore, z = y ust lie i M w Note: Projectio Theore The above theore is essetially the Projectio Theore Ideed, cosider the direct su H M Z M M, = = = + z, M, z M The vector M is called the orthogoal projectio of H o M z M We have z = i = M L- 8/8