If M is a closed subspace of a Hilbert space H then H %3DМӨМ'.Prove.

Q: Let M, N be closed linear subspaces of a Hilbert space H. If M 1 N, prove that M + N is closed.

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Q: Prove that every line through the origin in R3 is a subspace of R3.

A: To prove that every line through the origin in R3 is a subspace of R3.

Q: Let W be the set of all 2 × 2 matrices A = " 2 га b] such that Az = 0, where z = . Is W subspace of…

A: A set W of vector space V over field F is said to be subspace of vector space V if W is itself a…

Q: Let M be closed subspace of a Hilbert space H, and fix h e H. Then h has a unique representation as…

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Q: let S be the collection of vector [x] (3x1 matrix) in R3 that satisfy the given…

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Q: Verify that if W is a subspace of an inner product space V and v is in V, then perpw(v) is…

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Q: Let W be the set of all 2 × 2 matrices A = " d. such that Az = 0, where z = |. Is W subspace of M22?…

A: Use the properties of a subspace or definition of subspace. Assume that the condition is true and…

Q: If X is the subspace of `∞ consisting of all sequences of zeros and ones, what is the induced metric…

A: We have given that ,  X is the subspace of l∞ consisting of all sequences of zeros and ones. We need…

Q: can a subspace of R^n have a dimension less than n.

A: Yes , it can be .

Q: If W is a subspace of V and B is a hasis for V, then some subset of B is a basis for W. Select one:…

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Q: 2. Let W be an (n − 1)-dimensional subspace of V. Show that V has a basis B satisfying Bn W = Ø.

A: Please find the answer in next step

Q: Let Y be a closed subspace of a normed linear space X. Show that X is complete X are complete. Y AY…

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Q: Let W be the subspace of all symmetric matrices in M2,2. What is the dimension of W?

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Q: Let S be the subspace of R4 spanned by x1 = (1, 0,−2, 1)T and x2 = (0, 1, 3,−2)T . Find a basis for…

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Q: Let M be a closed linear subspace of a Banach space X. Prove that the quotient space X/M, with the…

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Q: Prove that the set of mxn upper triangular matrices is a subspace of Mmxn(F).

A: We have to prove that -

Q: Prove that if a 1-dimensional subspace W of Rn contains a nonzero vector with all nonnegative…

A: Let v be a nonzero vector with all non-negative entries in W and let d be the sum of all of its…

Q: Let S be a subset of a vector space V. Show that span (S) is the smallest subspace containing all…

A: Given that, S be a subset of the vector space V. We have to show that spanS is the smallest subspace…

Q: Any nonzero subspace W C R" has an orthonormal basis. True O False

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Q: Prove that the dual of l° is not isometriç to, 1', but contains a proper subspace that is isometric…

A: Consider that there are bounded functions on l∞ which are not of the form fy=∑xkyk for some x. Let,…

Q: If X is an inner product space, and A is subspace of X, then AL CAO no decision O ACALO A = A++O

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Q: Let E, be a closed subspace and E, be a finite dimensional subspace of a normed subspace X. Prove…

A: This is a question of Functional Analysis.

Q: If V is infinite-dimensional and S-is an infinite-dimensional subspace, must the dimension of V/S be…

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Q: Determine whether the set of all 3-tuples of the form (x, y, z), where z = x – 2y, form a subspace…

A: If W is a subset of a vector space V over a field F, then W is a subspace of V, only if it satisfies…

Q: In this set of exercises, H always denotes a Hilbert space. 1 If M is a closed subspace of H, prove…

A: Given: M is a closed subspace of a Hilbert space H.   To prove: M=M⊥⊥   To determine: If there is a…

Q: Prove that span (v1,v2,...,vn) is a valid subspace of R

A: We know that ,  spanv1 , v2 , .... , vn = a1v1 + a2v2 + ...... + anvn / a1 , a2 , ...... , an∈ℝ A…

Q: Find the orthogonal projection of onto the subspace W of R' spanned by -1 -1 -1 1 1 projw (7)

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Q: In this set of exercises, H always denotes a Hilbert space. If M is a closed subspace of H, prove…

A: Given that M is a closed subspace of H. To prove: M=M⊥⊥ .

Q: Prove that Co is a Banach space by showing that it is a closed subspace of l

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Q: () If S is a subspace in R that is spanned by then S is spanned by: O a. 1 and 0. 1 1 and 1 c. and…

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Q: If V is any three-dimensional subspace of R°, then V has infinitely many bases.

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Q: (b) Prove that if V is finite dimensional, then every subspace of V is also finite dimensional.…

A: (b)

Q: Verify that if W is a subspace of an inner product space Vand v is in V, then perpw(v) is orthogonal…

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Q: Let V = M22 and W = {invertible 2 x 2 matrices}. Determine whether W is a subspace of M2. %3D

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Q: If X is an inner product space, and A is subspace of X then A = A+ True False

A: Let A be a subspace of an inner product space X. Then (i) A⊥ is a subspace of X ; (ii) A∩ A⊥ = (0);…

Q: Show that an arbitrary linear combination of two states Vs in the vector space V with the same…

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Q: If M is linear subspace of hilbert space H then M is clos

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Q: Show that the set of all polynomials of degree at most 3 forms a subspace of Pn.

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Q: 12) Show that subset W of a vector space V is a subspace of V if and only if span(W) = W.

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Q: 1- Show whether the set of polynomials a, +at +at+at ,where a, =a, +a, is a subspace of P3 or not ?

A: Let V be a vector space over the field F. A non-empty subset S of the vector space V is said to be a…

Q: Prove that the dual space of 10 is not isometric to 11 but contains a subspace isometric to l1.

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Q: a) Prove that an operator P is the pr osed linear subspace M of a Hilbert

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Q: If X is an inner product space, and A is subspace of X then A = AL %3D O True O False

A: TRUE

Q: Let X be complete metric space, then a subspace M of X is itself complete iff the set M is closed in…

A: Hint: First suppose that M is compete then show that M is closed then suppose that M is closed and…

Q: If M‡ Ø is any subset of a Hilbert space H, show that M is the smallest closed subspace of H which…

A: Solution:Hilbert space H, M⊥⊥ smallest closed subspaceWe first show that for any M⊂H, M⊥is closed.…

Q: Find the orthogonal projection of onto the subspace W of Rª spanned by 1 projw (v)

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Q: 2. Let U and W be a subspaces of an n-dimensional vector space V and suppose further that UNW={0}…

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Q: Suppose V is a finite-dimensional inner product space Suppose U, W are subspaces of V . Prove that:…

A: Given: V is a finite dimensional inner product space and U,W are subspace of V. To prove :…