Give an example of a norm space.

Q: (a) Define the T1-space.

A: Define the T1 - space

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: If the dual X' of a normed linear space X is fininte dimensional, then X is finite dimensional

A: This is the question of Functional Analysis.

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: 6. Show that a compact metric space X is locally compact.

A: The given problem is related with metric space. We have to show that a compact metric space X is…

Q: Prove that the quotient of a locally connected topological space is locally connected.

A: We have to prove that Let q:X→Y be a quotient map, X is a locally connected space. Show that xY is…

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: Why do the set of all polynomials of degree n or less form a vector space, while the set of all…

A: Why do the set of all polynomials of degree n or less form a vector space, while the set of all…

Q: Prove that every well order is a linear order,

A: A set  is said to be well-ordered if it has a total order and has the property that every non-empty…

Q: Prove the set is a vector space by addition and scalar multiplication.

A: Let V be set, given as                                     V=x:x>0,x∈R elements of V be (a,b) and…

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: the quotient space of a T0−space necessarily T0

A:

Q: Define a compact space.

A: The objective is to give the definition of a compact space.

Q: Determine if the following statements are true or false. Briefly explain why. a) A quotient of a…

A:

Q: Define metric space and define its four properties

A:

Q: Every metric space is a hausdorff space .

A:

Q: Remark: If (V,) is an inner product space. Then for all uiv,wE Vand a,pEK " =« + Pp VW = a + p %3D

A: We have the inner product space V,.,. . A vector space V over a field F together with the map V,.,.…

Q: a. Is there any relation between reflexive normed space and a Banach Space? (If yes then prove) b.…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: Exercise. Every point in a discrete metric space forms an open ball.

A:

Q: Show that the norm of the maximum satisfies the definition of norm

A: The objective is to show that the norm of the maximum satisfies the definition of norm. Consider the…

Q: topological space a

A: Presumably, the closure of Q in R under the usual topology. It suffices to show that for every real…

Q: determine whether the set, together with the standard operations, is a vector space. If it is not,…

A: The given set is:0abc0def0

Q: Any cover of a metric space is open.

A: Any cover of a metric space is open_____?_____

Q: Show that ℓ^1 is a normed linear space.

A:

Q: Prove that any reflexive normed space is complete

A: A normed space X will be said to be complete if each Cauchy sequence in X is convergent in X.

Q: Find the dimensions of the following linear spaces. (a) R2x3 (b) The real linear space C6 (c) Ps

A: A matrix is given R2×3, It means this matrix have, No. of rows = 2, No. of column = 3.

Q: Find the null space for A. 1 2 -2 1 0 1 3 A = null(A) = span

A:

Q: Let [a, b] be an inter a space. Let ƒ:[a,b] → X

A:

Q: Show that if a Banach space X is non-trivial, X + {0}, 3). then X' # {0}.

A: According to the given information, it is required to show that:

Q: Prove or give counter example an infinite product of discrete spaces may not be discrete

A:

Q: Let x(bar) = (3,-5,2,1,-2) element of R5. Find the norm of x(bar) using the usual norm on R5

A: The norm of x is the distance between a point x to the origin.

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: Determine the dimension of the space spanned by the set

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

Q: Give two examples of finite dimension vector spaces.

A:

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

A:

Q: Show that R3 is a Banach space. Explain each step.

A: To find -

Q: What is Sierpinski Space? Also give example

A: Sierpinski Space is  a topological space. Now to define a topological space we have to define a…

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: Discuss Application of inner product spaces with examples.

A: Hi, since you have asked multiple questions, we will solve the first question for you. If want any…

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

A:

Q: Determine whether each product is an inner product space in R3. If not, explain why. 1) = u1v1 +…

A:

Q: what ordered pair is an example of a topological space that is connected but not path-connected,why?

A: We use contradiction method.

Q: Finite topological spaces are always complete. True False

A: Finite topological spaces are always complete. So it is true statement.

Q: The closure of a compact subject of a regular space is compact.

A:

Q: product space in R3. If not, explain why. 1) = u1v1u2v2u3v3

A:

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: 2.2 Is any discrete space complete? Give an example to substantiate your answer.

A: Yes, any discrete space is complete.

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.