(b) Let X C[0, 1], be the the space of continuous functions on the interval [0, 1]. %3D Consider X with the norm = max |x(1). re[0,1) Show that the subspace Y = (x € C[0, 1] : x(0) = 2.x(1)} is complete. %3!

Q: -4 Find the orthogonal projection ŷ of y = 4 onto the subspace 3 -3 W = Span { uj U2 = 2 Ex: 1.23

A: With the help of definition of projection of a vector on a subspace, we solve this problem.

Q: T X = R with standard topology { IneN} CR k= Show that the subspace topology on k is discrete.

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Q: 2. Let X be a normed space and UC X a closed subspace. The subspace U = {X E X' | Vu EU: X(u)=0} is…

A: Topic- Analysis

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Q: Consider L = I – A with A : X → X is a compact linear operator in a normed space X. Show that the…

A: The given problem is related with functional analysis. Given that L = I - A , where A : X → X is a…

Q: Let X be a normed space and Y C X a linear subspace. c) Show that Y is dense in X, if and only if…

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Q: (a) find the orthogonal complement S⊥, and (b) find the direct sum S⊕S⊥. S is the subspace of R3…

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Q: if a vector space is the set of real valued continuous functions over R , then show dy d y -9. dx?…

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Q: If V is the vector space of real valued continuous functions , then show that the set W of all d’y…

A: Our Aim is to show that the set W of all solutions of the differential equation3d2ydx2-5dydx+7y=0…

Q: Let W = {a + bx + cx² + dx°|c - 3d = 0} be a subspace of Pa. Then the dimension of W is equal to 3.…

A: PnF is set of all polynomials of degree nand smaller . PnF denotes the vector space over field F…

Q: Let W = (a + bx + cx² + dx*lc- 3d = 0}be a subspace of Py. Then the dimension of W is equal to None…

A: To find - Let W = a + bx + cx2 + dx3| c - 3d = 0 be a subspace of P3. Then the dimension of W is…

Q: -2 Find the orthogonal projection ŷ of y = 3 onto the subspace 2 -2 W = Span { u , U2 Ex: 1.23 : ||

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Q: In the vector space of all real-valued functions, find a basis for the subspace spanned by {sin t,…

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Q: et X = {(x,y, z): Vx² + y² + z² <1} be a subspace R° with the standard topology and 0 = (0,0,0).…

A: Given,               X=x, y, z:x2+y2+z2 ≤1  be a subspace of R3 with the standard topology and o=0,…

Q: Let W = {a + bx + cx² + dx³[c – 3d = 0 }) be a subspace of P3. Then the dimension of W is equal to O…

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Q: 1 -1 Find the distance from y to the subspace spanned by u. U= y= NO N1

A: Here given vectors are: u=2021y=1-11-1

Q: 6. Let C"(R) denote the sct of all real-valued functions defined on the real line that have a…

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Q: Let B = {1, x, eX, xe*} be a basis for a subspace W of the space of continuous functions, and let Dx…

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Q: {l: ol E M2x3: a = b = c = -3d } be a subspace of M2x3- Let W = d Then dimension of W is Оз 1 О4 O 2…

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Q: In the vector space of functions f: R → R, find a basis for the subspace spanned by {sin t, sin 2t ,…

A: Solving question by theorem vector space

Q: 11) Is the set W = {f(x) E P(F): f(x) = 0 or f(x)has a degree n} a subspace of P(F) if n > 1?…

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Q: 26. In the vector space of all real-valued functions, find a basis for the subspace spanned by {sin…

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Q: C0 ={x=(xn) : limn→∞ xn=0 } and la={x=(xn) : sup(xn)<∞ , a=∞} C0 is closed subspace of la space…

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Q: 6. Consider FV, where V consists of all real-valued functions. Let W C V, where W = {f € V | |f(x)|…

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: 4. Show/prove whether the following is a subspace. {x € R3:x =| 3 +s 2+t| 1 ,for s,t e R} 2

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Q: if a vector space is the set of real valued continuous functions over R , then show that the set W…

A: Given V is a vector space of all continuous functions over ℝ. W is a set of differential equation…

Q: 12. Let K C R" and let V = { E R" : a1y Vy E K}. Prove that V is a subspace of R".

A: According to our guidelines, we can answer only the first question, the rest can be reposted.

Q: Let W = {a + bx + cx2 + dx'|c-3d = 0} be a subspace of P, Then the dimension of W is equal to %3D 4.…

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Q: -3 Find the orthogonal projection ŷ of y = -1 onto the subspace -2 5 W = Span { u1 = -5 u2 = -3 Ex:…

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Q: Suppose that U cr|xyer F4 y EF, U is a subspace of F4- you don't have to! Find a subspace V of F4…

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Q: Is W = ((x,y) | xy=0} a subspace of R??

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Q: 4. Show/prove whether the following is a subspace. 1 {x € R3:x =| 3| +s 2+t1,for s,t e R}

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Q: Q.4 Show that U {(x, -x) |x in R} is a subspace of R?. %3D

A: Consider the given set. U=x,-x|xinR Let, consider that, f,g∈V And f, and g is defined as.…

Q: 5. Let C' (R) be the set of all continuously differentiable functions. Show that W = {f : f' = 4f}…

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Q: Is the set of all differentiable real-valued functions defined on R a subspace of C(R)? Justify your…

A: Given:Is the set of all differentiable real-valued functions defined on R a subspace of C(R)?

Q: 6 H 6 and let W be the set of all x in R³ such that u x = 0. (a) Show that W is a subspace of R³.…

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Q: Is the set of functions (taken from a vector space consisting of all functions of the real numbers):…

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Q: Example 3. The operator Ax = d/dt x(t), defined on the subspace M of C[0, 1] consisting of all…

A: consider the operator d/dt on C[0, 1] with domain consisting of continuously differentiable…

Q: 10. Let C' (R) be the set of all continuously differentiable functions. Show that W = {f : f' - Af =…

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Q: ) The degree of pa is equal to the dimension of the cyclic subspace Z (a ; T) .

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Q: Let W {a + bx + cx2 + dx³| a+b = 0 and c- 3d = 0} be a subspace of P3. %3D Then the dimension of W…

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Q: W e R"|2, > 0 over the field of reals is a subspace of the specified vector space ER"

A: Given that W=x1x2x3⋮xm∈Rm|x1≥0 and V=x1x2x3⋮xm∈Rm The objective is to show that W is subspace of V.…

Q: 2. The subspace (a) 2y x+y+z=0 1 in R3 is a plane whose equation is x-2=0 (b) y (c). x-2y+z=0 Py27.…

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Q: and uz = Find the closest point to y = in the subspace W spanned by u1 = The closest point is

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Q: Let B = {1, x, ex, xex} be a basis for a subspace W of the space of continuous functions, and let Dx…

A: Given: B=1, x, ex, x ex is a basis g=for a subspace W of the space of continuous functions. DX is…

Q: 1. Decide if the following subsets W C V are vector subspaces or not (justify) (a) W = x €RCR?. COs…

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Q: Is Q = {(x,y,z) | x=2y & z=1} a subspace of R?

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Q: 4) Suppose u, v E V, where V is an inner product space, and ||u|| = |||| = 1 and (u, v) = 1. Find u…

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Q: 2 Find the dimension of the subspace H of R spanned by – 12 8 ..... dim H =

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Q: I et f(x) = -3, g(x) = 3x – 9 and h(x) = -9x2. Consider the inner product (P. 9) = [ p(x)q(x) dx in…

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