Let V and W be normed vector spaces. Let us write VW to denote the vector...

 

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Let V and W be normed vector spaces. Let us write VW to denote the vector space of pairs with addition and scalar multiplication defined by the formulae VW = {(x,y) | x V,y W} (x1,y1)+(x2, y2) = (x1+x2, Y1 + y2) 1, 2 EV, 1, 2 W and a(x, y) = (ax, ay) TEV, y W, aF respectively. 1. Show that the vector space VW has a norm given by the formula |||(x, y)|| = ||x||v||y||w x V, yW where -v and ||-||w are the norms for the spaces V and W respectively. 2. Let (xn) and (yn) be sequences in the spaces V and W with norm limits x and y respectively. Show that the sequence (xn, yn) in VW converges in norm in VW to the limit (x, y). 3. Show that a sequence (n. Yn) in VW is a Cauchy sequence if and only if the sequences (an) and (yn) are both Cauchy sequences. 4. Suppose that V and W are Banach spaces. Is VW necessarily a Banach space? Prove it, or give a counterexample. 5. Let V be a normed vector space. Prove that addition +: VV V defines a continuous map. Let V and W be normed vector spaces. Let us write VW to denote the vector space of pairs with addition and scalar multiplication defined by the formulae VW = {(x,y) | x V,y W} (x1,y1)+(x2, y2) = (x1+x2, Y1 + y2) 1, 2 EV, 1, 2 W and a(x, y) = (ax, ay) TEV, y W, aF respectively. 1. Show that the vector space VW has a norm given by the formula |||(x, y)|| = ||x||v||y||w x V, yW where -v and ||-||w are the norms for the spaces V and W respectively. 2. Let (xn) and (yn) be sequences in the spaces V and W with norm limits x and y respectively. Show that the sequence (xn, yn) in VW converges in norm in VW to the limit (x, y). 3. Show that a sequence (n. Yn) in VW is a Cauchy sequence if and only if the sequences (an) and (yn) are both Cauchy sequences. 4. Suppose that V and W are Banach spaces. Is VW necessarily a Banach space? Prove it, or give a counterexample. 5. Let V be a normed vector space. Prove that addition +: VV V defines a continuous map.