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.
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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