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Let V be a vector space: d) Suppose that V is finite-dimensional, and let S be...

Let V be a vector space: d) Suppose that V is finite-dimensional, and let S be a set of inner products on V that is (when viewed as a subset of B(V)) linearly independent. Prove that S must be finite

e) Exhibit an infinite linearly independent set of inner products on R(x), the vector space of all polynomials with real coefficients.

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d) Since V is finite dimensional implies , the dual of V is also finite dimensional and since S be a subset of , which is linearly independent implies S is a linearly independent subset of a finite dimensional vector space . Hence finite.

e) Consider the set . We claim that the given set is linearly independent. Note that if the set is linearly dependent then there exists a natural number n real numbers , i=0,...,n, such that , then consider the polynomial , this this polynomial has more than n roots. Which contradicts the fundamental theorem of algebra. Hence the given set is linearly independent.

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