Prove that any linear map between finite dimensional normed linear spaces is continuous.

Q: Show that [0, ∞) and (0, 0) ad subspaces of R with the usual topology are not homeomorphic.

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Q: Let Y be a vector subspace of a normed space X. Show that the closure of Y is also a vector…

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Q: show that the discrete metric on u Vector space X+3 of eannot be obtained from a norm.

A: To Show: Show that the discrete metric on a vector space X ≠ 0 cannot be obtained from a norm.

Q: 3. Show that closed subspace of a complete metric space is complete. 4. Let f be a continuous…

A: As per norms, the first question is answered: To prove that a closed subspace Y of a complete metric…

Q: Prove that linear maps are bounded for multivariable calculus

A: In this question, concept of linear map is applied. Linear Map The matrix of the linear map can…

Q: Prove that any Space is complete. finite dimensional normed

A: Introduction: Completeness is a property of a topological space. When in a topological space, every…

Q: Every continuous function from normed space X into normed space Y is bounded linear operator True…

A: True

Q: Define diffeomorphish between two manifolds and Completly explain why any linear map fiM_N. between…

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Q: If X, is a dense subspace of a normed linear space X show that X X'.

A: We will use basic knowledge of dual of normed linear spaces and the concept of isometric…

Q: Prove that coo cannot be a Banach space under any norm.

A: To prove: c00 cannot be a Banach space under any norm.

Q: normed lineer spare X is finite demisional Linear transformation. X is bounded om If them every

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Q: 6. Show that the closure Ỹ of a subspace Y of a normed space X is again a vector subspace.

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Q: Prove that an affine map is continuous if and only if its linear part is continuous

A: If T:A→Rk is the affine map, where A is the affine span of an affine independent subset of Rn. Let…

Q: Let X be a normed space and Y be a closed subspace of X. Define the quotient norm on the space X/Y.…

A: Definition : Let X be a linear space over the field K. A function ||•||: X to R is said to be norm…

Q: Show that the element 0 in a vector space is unique

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Q: Define quotient space of a vector space.

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Q: Prove that a vector space is infinite-dimensional if and only if it contains an infinite linearly…

A: Given the statement that a vector space is infinite-dimensional if and only if it contains an…

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 F, G be linear maps of vector space V onto itself. Show that: (FoG)-1 = G¬1 oF-1.

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Q: If X a normed space and y is a proper proper Linear subspace of X. show that y has empty interior.

A: We proved by contradiction. Suppose Y has a nonempty interior then Y contains a open ball. Then we…

Q: Show that V={(x,2020x): x in R} is a vector space

A: According to the given information, it is required to show that the given set is a vector space.

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: Theorem: Every normed vector space is a metric space but the converse is not true in general Proof

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Q: snip

A: Theorem:                  Suppose <v,w> is an inner product on a vector space V.  Then…

Q: Show that if a non-zero element in a normed space X, then there is a bounded linear functional f on…

A: We need to show that if a non-zero element in a normed space X, then there is a bounded linear…

Q: Let X and Y be compact spaces. Then X × Y is compact.

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Q: Show that a subspace of a Hausdorff space is Hausdorff.

A: Prove that, a subspace of a Hausdorff space is Hausdorff.

Q: Prove that a linear map on a normed vector space is bounded if and only if it is continuous.

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Q: Show that a finite union of compact subspaces of a topological spaces X is compact-

A: We have to show that the finite union of two compact subspaces of a topological space X is compact.

Q: / Let (X₁T) and (Y₁J) be top. Spaces and fix-x be a surjective and continuous map, then y is Compact…

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Q: xamples -X = R with the norm ||x|| = |x| x € R is normed space

A: Here we needs to show real number is a normed space under modules.

Q: Show that ℝ2 is a vector space with usual addition and multiplication.

A: To prove that R2 is vector space we have to prove following; 1. R2 is abelian group under addition.…

Q: If them every normed lineers spare X is finite demisional Linear transformation. X is bounded a

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Q: Prove the locally compact normal linear space is finite dimensions.

A: We have to prove the  locally compact normal linear space is finite dimensions First we define the…

Q: Prove that a closed linear subspace of a reflexive space is reflexive.

A: This is a problem of functional analysis.

Q: The vector space C[a, b] of all continuous real-valued functions on [a, b] is not the span of any…

A: A vector space is a set whose elements, often called vectors, may be added together and multiplied…

Q: a discrete space

A: We know that in a discrete topological space X, every point in X is an open set.

Q: prove that any finite dimensional normed space is reflexive.

A: Let us suppose that we have a finite-dimensional normed space, X For any natural number, n, let…

Q: Prove that a finite dimensional normed linear space is always reflexive.

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Q: Is it true that if two vector spaces V and W both have dimension 2022, then they must be isomorphic?…

A: Suppose V and W are two vector spaces. We know that , V and W are isomorphic to each other if there…

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: Prove that a normed linear space is complete if its unit sphere is complete.

A: We prove that a normed linear space is complete if its unit sphere is complete. Its straightforward…

Q: Let X be a normed linear space such that its dual X' is separable, then show that X is separable ?…

A: This is a problem of Functional Analysis.

Q: Show that a space X is Hausdorff if and only if the diagonal {(x, x)|x € X} is closed in X2.

A: We use basic topological definition to prove this.

Q: Let X, Y be normed linear spaces. On X x Y define || (x,y) |l, = || × |lx + || y lly and + || y ily…

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Q: not vector spaces

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Q: A subspace of a completely regular space is completely regular.

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Q: Show that if W is a subspace of a finite-dimensional vec- tor space V, then W is finite-dimensional…

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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 :…