Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

Q: (1.) Prove that (R", d) with d : R" × R" → R defined by d(r, y) = E(Ik – yk)² k=1 is a metric space.…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Let (X, M, µ) be a measure space such that uX = 1, and let f : X → (0, 1). Show that u) / fdµ 2 1

A:

Q: If X =R and d(x.y)30, then X metric space not metric semi metric

A:

Q: Let (X, d) be a metric space and let A, B⊆X be such that A is connected, and A∩B ≠ ∅ and A∩ (X − B)…

A:

Q: Let (X,d) be a metric space and E ⊆ X. Prove that if E is compact, then E is bounded

A:

Q: Let V, W be finite-dimensional inner product spaces and T € L(V, W). Prove: (a) If T is injective,…

A: Injective Function:  A function is said to be injective if each element of the codomain is mapped to…

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

A:

Q: Suppose V is an inner-product space and u, v e V satisfy ||u|| = ||v|| = 1 and (u, v) = 1. Prove…

A:

Q: b) Let X be an inner product space and x, y e x.Prove that xly if and only if kx + y = |lk | + |y*k…

A: Given that X be an inner product space and x,y∈X Let, x⊥y. The objective is to prove that…

Q: E be a Euclidean space of dimension (E(1, f)) = dim(Ker (f- id)), then s minimal.

A:

Q: Let (X,d) be a metric space. For x,y in X define e(x,y)=min{1,d(x,y)}. Prove that (X,e) is also a…

A:

Q: For every x, y E R, let p(x, y) = |x² – y²\. Is p a metric on R?

A: Hello. Since your question has multiple parts, we will solve first question for you. If you want…

Q: Let (X, d) be a metric space. Show that the function f: X X x X defined by f(x) = (x, x) is…

A:

Q: 1 Show that IS the separable me tre space (R, d), where el(x, y) = be-y/

A: Here we use basic real analysis

Q: Let X be a normed space, Xn, Yn EX such that Xnx, Yny, then 1-Xn + Yn → 4- ||xn-ynll→→llx - yll.…

A: Given that X is a normed space, xn, yn ∈X such that xn→x, yn→y. To find1. xn+ yn→x+y 2. λxn→λx ∀λ∈F…

Q: Let X be a normed space then 1- ||0|| = 0 2-I1-xl = |지l, Vx E X 3- ||x - yl| = |ly - x|,Vx, y E X…

A: As per the policy, we are solving first three subparts. Please resubmit the question and specify…

Q: Let X and Y be normed linear spaces, and T E B(X,Y). Prove that 38 > 0 such that 8||x|| < ||T(æ)||,…

A: Consider the given information.

Q: Let (X,d) be a metric space. Prove that the function d' :X x X →R given by d (x, y)= min {1,d(x, y)}…

A:

Q: View S' as the boundary of the 2-disk D² = {(x, y) E R² | x² + y? < 1}. Fix To E S', and let X = (S'…

A: Given : D2=x,y∈ℝ2|x2+y2≤1 where x0 belongs to S1 which the boundary of 2-disk. X=(S1×S1)∪(D2×x0) To…

Q: Let f be a continuous mapping of a metric space X into metric space Y,g be a continuous mapping of…

A:

Q: Let (x, g) be a metric space. Is the function d (x, y) = k.g (x, y) (k € R) defined from XxX to R a…

A: Let X,g be a metric space . Given that function X×X to R defined as  dx,y=k·gx,y where k∈R A metric…

Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

A:

Q: å:RR→R be defined by dcx.y)=|xl+1yl, HxixR (R,d)is Than a metric space.

A: As per our guidelines we supposed to solve the very first question when the multiple questions are…

Q: Let (X, d) be metric and PXIR space, dosy) I+ dexy) a de fined by qusN pusy) = ニ show that p is…

A:

Q: Let X = Rand de f ine ||| on X as ||r|| = Va then X is normed linear space

A:

Q: ) Let X be a finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

A: Sufficient to prove that M is closed and bounded

Q: let X be a firite dimensional notm Space Then Prove that is compat.

A: As X is a finite dimensional norm space, this means that, dimX is finite.

Q: ht x and y be normed spaces. Show that a Lincar operator T:Xy is bounded If and only T maps bounded…

A:

Q: Let V be a real inner product space, and let u, v, w E V. If (u, v) = 1 and (v, w) = 3, what is (3u…

A:

Q: Let (X, E, µ) be a measure space. Show that f is integrable if and only if fx ]f]dµ < o∞. Moreover…

A:

Q: 11. Let (V.||-|) be a normed vector space. Suppose e : V - R has the property that e(r+y) =…

A:

Q: i) Pe(x) = 0 iff r € Ë. ii) Prove that pe is uniformly continuous.

A: i Let E is the non-empty subset of a metric space X,d the distance from x to E is defined as…

Q: Let V be a finite-dimensional inner product space over R. Given u, v E V, prove that ||u + v||? –…

A:

Q: Give the definition of a separable metric space (S, d). Show that the metric space (R, d), where…

A:

Q: Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = &lt;x, y&gt;z for all x…

A:

Q: ) For the vector space C[0,1], define the inner product (f,g) = , f(x)g(x)dx. Let f = x + 1 and g =…

A:

Q: Define on R, the metric d defined by d(x, y) = |1+1리 1+ |y| Then (R, d) is not complete. True False

A: Given

Q: Theorem 2: Let X be a finite dimensional normed space and let r > 0. Then, the closed unit ball B[0;…

A: Theorems and definitions we are going to use: 1. Definition: (X,|| . ||) is said to be bounded if…

Q: Consider the linear space C[0, 1] equipped with the norm ||f || = max |f(t)|. te[0,1] Prove that…

A: Given: The linear space C0,1 equipped with the norm, f=maxt∈0,1ft   To prove: There is no inner…

Q: Let X,d) be a metrie space. Define d': Xx X >R by: d'Cxoy) Show that (X.d') is min (dl8), 1). a…

A:

Q: Let E be a Euclidean space with inner ratic form. For all u, v E E, we have the C y(u, v)² ≤…

A:

Q: H.W.2 Exercise Prove that: 1. (IR, J- |) is a complete metric space. 2. (Q. |-1) is not complete…

A:

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

A:

Q: the usual metric space defined by  d(x,y)= x-y prove the four propertis of metric space

A: Applying four properties to this metric

Q: The set R2 is a vector space over R. How Prove it ?

A: We need to check each and every axiom of a vector space to know that it is in fact a vector…

Q: Use Green's Theorem to evaluate S. axy dy + cxy dy, where C is a counterclockwise oriented…

A: Given : a=2, c=8 We have to use Green's Theorem to evaluate            ∫C axy dy+cxy dy where C is a…

Q: Let R have its usual topology and let Y = {x, y, z}. Define f: R → Y by: %3D x 0 (х, f(x) {y, (z,…

A: Quotient topology: If X is a topological space and A is set and if p:X→A is a surjective map, then…

Q: An Elliptical norm for R2 is defined by + 4u2, where x = (1,u). (i) Prove that l is a norm for R2…

A: This is a problem of Functional Analysis.

Q: Give the definition of a separable metric space (S, d). Show that the metric space (IR, d), where…

A:

Q: Prove that any smooth manifold (M, A) has a countable smooth atlas V (so that D(V) = A)

A:

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.