Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric                      d((x1, x2), (y1, y2)) = max{d1(x1, y1), d2(x2, y2)} is also separable space.

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y.

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Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. *

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

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Q: (b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ?…

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

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Q: Consider ℝ3 with the metric d((x1, x2, x3), (y1, y2, y3) = |x1 - y1| + |x2 - y2| + |x3 - y3|. i.…

A: Given metric is          d((x1, x2, x3), (y1, y2, y3)) = |x1 - y1| + |x2 - y2| + |x3 - y3|.   For…

Q: 5. Let V be an inner product space, and T E L(V). Prove that null(T) = (Im(T*)). %3D

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Q: On a previous homework, you proved a Bolzano-Weirstrass theorem for ℝ3 with the metric d((x1, x2,…

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…

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A: Consider the given information.

Q: On a previous homework, you proved a Bolzano-Weirstrass theorem for R3 with the metric d((x1, X2,…

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Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

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Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. * O…

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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: The curl of f(x,y,z)=xyzj is None of these. хуk-yzi О ху-yzk ООО О

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Q: 12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3D

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Q: Prove the parallelogram law on an inner product space V; that is, showthatllx + Yll2 + llx - Yll2 =…

A: Prove the parallelogram law on an inner product space V; that is,  Show that…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.