Let Y be a closed subspace of a normed linear space X. Show that X is complete X are complete. Y AY and ---

Q: If V = R is a vector space and let Hbe a subset of V and is defined as H={(a,b,c): a² +b² = 0,c>0}.…

A: To show that H is not a subspace of a vector

Q: Prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and, whenever a ∈…

A: We need to prove that a subset W of a vector space V is a subspace of V if and only if W = ∅, and,…

Q: Let (X, F) be a space. Prove that a subspace of a subspace is a subspace of (X, F).

A: Here given X, I is a topological space.

Q: If U is a proper subspace of a finite-dimensional vector space V,show that the dimension of U is…

A: We will prove this by taking help from two theorems. Please see the subsequent steps:

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

A:

Q: Let U and W be subspaces of V over a ficld F. Show that the union U UW is a subspace if and only if…

A:

Q: Suppose U1,U;, -..Um are finite-dimensional subspaces of V. Prove that U + Uz + .. +Um is…

A: Suppose U1, U2, ... , Um are finite-dimensional subspaces of V. To prove U1+U2+ ... +Um is finite…

Q: 6. Let U and W be 3-dimensional subspaces of a 4-dimensional vector space V, and suppose that U + W.…

A:

Q: Let A and B be disjoint compact subspaces of the Hausdorff space X. Show that there exist disjoint…

A:

Q: Show if W is a subspace of the vector space X, then W is convex.

A: Definition used: A set K in a vector space V is called convex if and only if  x, y belongs to K,…

Q: If V is a finite dimensional vector space and W is a subspace, the W is finite dimensional. Prove it

A: NOTE:  If V is a finite dimensional vector space and W is a subspace of V, then W is finite…

Q: Let V be an n-dimensional vector space with inner product ( , ). Let U and W be subspaces of V such…

A: Let V be an finite dimensional inner product space and U and W be subspace of V and U∩W=0¯ We have…

Q: Let V and W be subspaces of R", and dim V and dim W are strictly less than n. Assume that no…

A: Given:- V and W are subspaces of ℝn              dim V<n, dim W<n              Also, there is…

Q: Let H and G be subspaces of a vector space V. Show that HnG is a subspace of V.

A: We have to show that intersection of two subspaces is a subspace.

Q: Let V be a finite-dimensional vector space, and let U, W be subspaces of V. Prove that (U + W)° C…

A:

Q: show that if w is a subspace of a finite - dimensional vector space V and dim (w)= dim (V) then W =…

A: In this question, we will use the fact that If dim(W) = dim(V ) = n, then a basis for W is a…

Q: Let W1 and W2 be subspaces of a finite-dimensional vector space V. Determine necessary and…

A: Consider W1 and W2 be the subspaces of finite dimensional vector spaces V.

Q: Let U and W denote subspaces of Rn, and assume that U ⊆ W. If dim U = n − 1, show that either W = Rn…

A:

Q: Determine the dimension of the subspace of R consisting of all XeR* such that x1 + 2x, = 0 and X3 –…

A:

Q: Let W be a subspace spanned by the u's, an

A: Given, W be a subspace spanned by u's, and write y as the sum of a vector in W and a vector…

Q: let H= { a b 0 \ a ∈ R and b∈ R.} Show that H is a subspce of R3 by showing that it…

A:

Q: snip

A: We have vector space of all real numbers over real numbers And S is subspace of vector space given…

Q: is

A:

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: Let W is a finite dimensional subspace of an inner product space V and y is any vector in V. The…

A: Let W be a finite dimensional subspace of an inner product space V. Let T be an orthogonal…

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: If y is in a subspace W, then y - ŷ is not zero. Select one: O True O False

A:

Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

A:

Q: Let V be a finite dimensional vector space and W CV a subspace. Prove that W has a complimentary…

A:

Q: Show that a subset W of a vector space V is a subspace of V if and only if span[W] = W.

A:

Q: If S is a set of vectors in an inner product space V(F) then prove that S- is a subspace of V.

A:

Q: Let W be the set of all the vectors of the form y where x and y are any real numbers. 2х-Зу…

A:

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

A: To show that W is a subspace of V if and only if span (W) is equal to W.Let W be the subspace of V.

Q: If x is in W, then x is orthogonal to some vector in a subspace W

A: SOLUTION-

Q: Let T and T' be topologies on a set X such that T' is strictly finer than T. If Y is a subset of X,…

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

Q: Let T be a linear operator on the space V, and let W, ,W hecessary and sufficient condition that…

A:

Q: snip snip

A: Consider the set S=x,y| xy⩾0. Note that 1, 2,  -2, -1∈S. Because it satisfies 1·2≥0   and   -2·-1≥0.…

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

A:

Q: Let V be a finite-dimensional vector space and let U be a subspace of V. Show that if Lim(U) =…

A:

Q: Let Wi and W2 be two two-dimensional subspaces of the linear space R³, that is dim W1 W = W1n W2 is…

A: Given W1 and W2be the two-dimensional subspace of linear space ℝ3. And dim W1=dim W2=2 let  x∈W1∩W2…

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

A:

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

A: This question is related to vector space . Solution of this question is

Q: Let V be a vector space and let S = {v,,…, Vn} be a set of vectors in V. Prove ... that the span of…

A:

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: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

Q: Let W be a subspace of V, and suppose that u1, u2, Uz E W are three linearly indenendent Yectors…

A:

Q: Let V and W be subspaces of R", and dim V and dim W are strictly less than n. Assume that no…

A: To prove: If V and W are subspaces of ℝn such that dim V < n, dim W < n, V is not contained in…

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

A:

Q: Let H be a subspace of real space R. In terms of a geometrical definition of H, which of the…

A: Given: