Show that the space (P is complete.

Q: Show that every second countable (A2-space) space is separable.

A:

Q: (a) Define the T1-space.

A: Define the T1 - space

Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

Q: Is the euclidean space R³ is a complete metric space

A: We have to check whether euclidean space R³ is a complete metric space or not .

Q: Consider the collection of all points of the form – 2w] 3w w Express this collection as the span of…

A:

Q: Suppose A and B be subsets of metric space. Show that if A is compact and B is closed then A + B is…

A: Given A and B are subsets of a metric space. and also Given A is compact , and B is closed. We have…

Q: Show that R is a Banach space.

A: Banach space: A complete normed linear space is called Banach space. Given: ℝ3=x(1, x(2), x(3)) :…

Q: Show with all the relevant details, albeit succinctly that every compact metric space M is complete.

A: Here we use sequential definition of compactness so that we use that in proving the completeness of…

Q: (2) Find all connected covering spaces of RP2 v RP².

A:

Q: a) Show that every closed subset of a compact space is compact but the converse need not to be true…

A: As per the guidelines I had given the proof of the first theorem. Kindly repost the another theorem…

Q: be a closed subset of space Xa

A:

Q: ExeraiSE 4Let the Open set dEnse 5- YOVE metrc Space Cs,d) Hen in

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Q: A ° = A – JA

A:

Q: Let X be a topological space

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Q: Let A be a closed bounded subset of a metric space (X, d). Is A a compact subset? Explain why.

A:

Q: Every metric space is a hausdorff space .

A:

Q: Suppose H is an inner |(x,y)| ≤ 11x11-1111 product space. Show Ужуєн.

A: Given: H is an inner product space. To prove: x,y≤x·y, ∀x,y∈H

Q: 1. (a) Define what a linear subset of a projective space is.

A: As per bartleby guidelines for more than one questions asked only first should be answered. Please…

Q: Every bounded domain subset of D containing

A: Set theory

Q: Prove that the space of all real numbers is complete

A:

Q: Show that R is a bianch space.

A:

Q: Let ?1and ?2 be compact subsets of a metric space X. Show ?1 ∪ ?2 is compact

A:

Q: (a) Prove that the metric space (2,11) is complete,

A: Complete Metric Spaces:- A Metric Space (X,d) is said to be complete iff every Cauchy sequence in X…

Q: LBe M, N subsets of space W. Then span (MUN) = span (M) + span (N)“ %3D

A:

Q: Let (X, M, u) be a measure space wE

A: Given, (X, Μ, μ) be a measure space where μ is a probability measure. Assume thatf, g : X→ℝ are…

Q: Let (X, d) be a complete metric space and let F ⊆ X be a closed set. Show that F is complete.

A:

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

A: Given: A is 3*3 matrix with rank A = 2

Q: Q7: Prove that ( IR", ||·|| ) is complete metric space.

A:

Q: every finite subset of metric space is compact. ture or false?

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Q: A of a metric space (x, d.)

A:

Q: Prove that (R", II-|) is complete metric space.

A: This is a problem of Functional Analysis.

Q: Determine the dimension of the space spanned by the set

A: Given,               S=12-13, 1002, 28-48, 1111, 3306

Q: Q7: Prove that (R", ||-|| ) is complete metric space.

A:

Q: ovide an illustration of the statement "Every normal space need not be a ular space."

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Q: 3: Prove that any reflexive normed space is complete.

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Q: Every closed subset of a connected metric space is connected. O True O False

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Q: Let M# ¢ is subset of a Hil bert TT H. Show that M is the Smallest Space closed Space. of H which…

A:

Q: Let N and N' be normed linear spaces with the

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Q: Prove that if X,d is a metric space and G ⊆ H ⊆ X, then G-bar ⊆ H-bar.

A:

Q: (a) Show that every metric space is T2-space.

A:

Q: Q7: Prove that ( IR", II-||) is complete metric space.

A:

Q: Every inner product space is a metric space O True O False

A: TRUE

Q: Every closed subset of a connected metric space is connected. True False

A:

Q: Show that if X and Y Hausdorff spaces, then so is the product space X × Y. are

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Q: (a) Prove that the metric space (Z,) is complete.

A: Complete Metric Spaces: A metric space (X,d) is said to be complete iff every Cauchy sequence in X…