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10 Nov 2019

2. For a metric space (X, d), boundary of a set A is defined as bdA = clA ∩ cl(X − A). Prove each of the following for a set A:

(i) The bdA is a closed set.
(ii) A is closed if and only if bdA ⊂ A.

(iii) clA = Ao ∪ bdA.

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Keith Leannon
Keith LeannonLv2
28 Oct 2019

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(i) Give an example of a vector space V and two subspaces W_1 and W_1 of V such that W_1 ∪ W_1 is not a subspace of V .

(ii) If W_1 and W_1 are subspaces of V , prove that W_1 ∪ W_2 is a subspace of V if and only if W_1 ⊆ W_2 or W_2 ⊆ W_1.

please i need help.

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Let P3 be the vector space of all polynomials with real coefficients of degree at most 3.

a) Show that the set E={p(x) ∈ P3 : p(0)=p(1)=0} is a subspace of P3.

b) Show that if p ∈ E, then p can be expressed in the following form:

p(x)=x(x-1)(ax+b) : a,b ∈ R

Hence, find a basis for E and dim(E)

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