2. Let X be a normed space. For each element (x, y) = XX, put ||(x, y) || 1 max(x, y). Show that || ||1 and || || and ||(x, y) || : ΧΘ Χ. 2 =: Show that if (a) is convergo coquongo in 0 =: then it is also are equivalent norms on |||| + ||x || nuorgo OBUOLI903

Q: A: For each of the following maps T: V→ W, find the spaces V and W and say if they are linear or not…

A: The given map T:V→W such that Tx, y=1, y. We have to check whether the map is linear or not.

Q: Let (X, d) be a metric space, prove that for any x, y, z E X, we have |d(y, z) - d(x, z)| ≤d(x, y).

A:

Q: Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 -…

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: (N2) (Homogeneity) ||ax || = |a| ||x || for all a eR %3D

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

Q: Let x = (13, −20, 15), v1 = (1, −2, 3), and v2 = (−1, 1, 1). Then the set {v1, v2} is orthogonal. It…

A:

Q: x · y = (x + y² – x² - y²)

A:

Q: 1. Let V be an inner product space. Suppose f,g E V such that f -gg (a) Prove that 1I2 = |lf- gll2 +…

A: To prove the required norm property in an inner product space, under the given conditions

Q: The region bounded by f(x) 2x² + 12x + 14, x = = 0 is rotated about the y-axis. Find the volume of…

A:

Q: a normed space, mapping such that ker(T) = {0}). Show that: defines a norm on E. any vector space…

A: Answer given below.

Q: 4. Show that the norm on l0 given by I| (11, .., In...) |lo= sup{|r;| : j= 1,2, ...} can not be…

A:

Q: 3. Let V be a finite dimensional vector space and let (.,.,) be an inner product on V. Suppose that…

A:

Q: - Which of the following formulas define norms on R²? (Let (x, y) be the co- ordinates in R².) (a)…

A:

Q: Let v,w ɛ R" and define A = vw². Prove that ||A||2 result true for the Frobenius norm, i.e. is…

A: Let v,w∈ℝn define A=vwT. v=x1,x2,...,xn and w=y1,y2...,yn. Then,…

Q: Define a metric as d(x,y)=|x1-y1|+|x2-y2|. Find the set of all points x=(x1,x2) in R2 such that…

A: