5. Let p = i o projw : R" → W → R" be the function defined by orthogonal projection onto W followed by inclusion, i.e. i(x) = x for all xe W. Show that the null space of Null(p) := {v € R" : p(v) = 0}

Q: 8.0. If x-(x,, X, ...,%)«R', define [xL by Prove that xJkL is a norm on R'. N I|} dns - x]

A: We will solve the first question highlighted in yellow. If you want the second question to be…

Q: Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norn of function f(x)…

A: This is a problem of functional analysis.

Q: 1) Let f(z,y)- . Compute DS(z,y) where u- (4-9)

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Q: 10. Let X be an inner product space and T: X→→→→→X an isometric linear operator. If dim X<∞, show…

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Q: Suppose {u1, Span{u;}nSpan{w;}={0}. Ur, W1, Ws} is a linearly independent subset of V. Show that •..…

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Q: 6. Consider a normed vector space S = (R,|- |). Show that, for all r, y e S, |z - y| 2 | – l9|-

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Q: 5. Let V be an inner product space, and T E L(V). Prove that null(T) = (Im(T*)). %3D

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Q: Suppose V is an inner-product space and u, v e V satisfy ||u|| = ||v|| = 1 and (u, v) = 1. Prove…

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Q: b) Let X be an inner product space and x, y e x.Prove that xly if and only if kx + y = |lk | + |y*k…

A: Given that X be an inner product space and x,y∈X Let, x⊥y. The objective is to prove that…

Q: Date. AC) Lat normed linear space X, Y= spomE and aex only if fla) = 0 wheneres f EX' omd f30…

A: To show the necessary part- To show that for each f∈X' whenever f vanishes on E, i.e.if f=0 then…

Q: 8.F. Ifx= (1, I, --- ,)€R", define x, by e that z- l, is a norm on R'. Prove

A: Given that x=x1,x2,..,xn and x1=x1+x2+...+xn. The objective is to find the whether x1=x1+x2+...+xn…

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Q: show that the integraloperatort (Tx) (+)= Skctsasx Luldewin Linear and maps c[ol] +o s [osl] and…

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Q: 5. Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norm of function…

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Q: 4:x→X (when X is a normed Space) is continous at xo € X iff for any Sequeue ie AS n s

A: Let f:x→X (when X is a normed space) is continuous at x0 ∈X iff for any sequence  xn ∈X, xn→х0  ⇒…

Q: dz Let z-ay" – 4r, * = uv and y-v - 3u. Compute Ən (u,r)-(1,1) -

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Q: 6. Consider S = [a, b] with p(x, y) = |x – y|. Verify that T : S – S with T(x) = - and a > 1 is a…

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Q: & 10 Let 20xx,y)=x² be a function defined by on al disk I in the positive quadrant containing the…

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Q: 3. Use Green's Theorem to evaluate G³ï+xj)•dr around the perimeter of the rectangle 0<x<2,0<y<3.…

A: Given ∫C(y2i+xj).dr We know that dr = dxi+dyj Thus ∫C(y2i+xj).dr = ∫C(y2i+xj).(dxi+dyj)  =…

Q: Let A: C [0,1]→C [0,1]. Prove that operator continuous reversible, find A. 2.11. Ax(t)= x(1)- -…

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: 3. lind Analytic function in, the disk 7z-11<1 whose veal part

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Q: 9. Show that a bounded linear operator T: H–→H on a Hilbert space H has a finite dimensional range…

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Q: ) Let X be a finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

A: Sufficient to prove that M is closed and bounded

Q: ht x and y be normed spaces. Show that a Lincar operator T:Xy is bounded If and only T maps bounded…

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Q: 9. Let be a bounded open subset of C, and 4: Prove that if there exists a point zo E such that 2 a…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: 28. Bounded by the planes z = x, y = x, x + y = 2, and z

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Q: Show that there does not exist a conformal map from C onto the region {z : [2| > 1}.

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Q: 0:1 Show that norm ||x|| = (Elx¡l?)ž satisfies the conditions of normed space. %3D

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Q: 6. Let y = 9 , U1 , U2 1 |3| L10] 1 Compute the orthogonal projection of y onto W = Span{u1, u2}.…

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Q: 2. Use Wronskian to show that fi = x, f2 - Inz, and fa C (R). z Inz are linearly independent in

A: First we will calculate the wronskian of given functions f_1,f_2,f_3

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Q: The main point of this exercise is to use Green’s Theorem to deduce a special case of the change of…

A: Please note that although it is written in your question post Solve all of them please, but these…

Q: 4 Let y = U2 = -7 U1 = -4 4 -6 -2 -9 Compute the distance d from y to the plane in IR° spanned by u̟…

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Q: If X is finite dimensional hor med space o T: X-Dy is a Linear operator from X to normed space y.…

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Q: Are the following sets of functions, each defined on x € R, linearly independent? Use Wronskian. (a)…

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Q: 4) Suppose that T is an operator on an inner-product space V such that T2 = T. Prove that T is an…

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Q: Let C be the rectangle in the xy-plane with vertices (0,0), (1,0), (0, 2) and (1, 2), and let F(r,…

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Q: 2. Show using only the Cauchy-Schwarz inequality that if u,vE R", then || u- v ISl u || +| v | (1

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Q: Show that, with z = Rw, for some bounded w, || z(jw)|| w llw(jw)| ||R||. = sup means that -

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Q: Consider the linear space C[0, 1] equipped with the norm ||f || = max |f(t)|. te[0,1] Prove that…

A: Given: The linear space C0,1 equipped with the norm, f=maxt∈0,1ft   To prove: There is no inner…

Q: Q3) Let L = Z* and |x ||= max {]x1 - x3l, x2l, 15x,1} Vx = (x1, X2, X3, X4) E Z*. Is (Z*, ) is a…

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Q: 10. If A is 3 x 3 with rank A 2, show that the dimension of the null space of A is 1.

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Q: 12) Prove Ñ' (t) are Smooth space for any always orthogonal (*+€ dom Ŕ³). that Curve R(t), 7(t) and

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Q: Let X = (!,2) and T3} U = (1,2-)inel,2,U+ jult. %3D n+1 show that X is not T - space

A: This is a problem of Topology.

Q: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

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Q: 2 - Find the inter val on which f are Con Cave down and infleobion Points a -y = x2 5X+10 6- ソ= (¥+…

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Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is 1.

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Q: Suppose that U is a linear transformation from R" into R" that is isometric, meaning that ||Ux|| =…

A: Let U: Rn → Rm be the isometric transformation and Ux=x for all x∈Rn (a) We have to prove that…