3. Let (V, ) be an F inner product space. Let u, v E V such...

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3. Let (V, ) be an F inner product space. Let u, v E V such that < x, u >= < x,v > for all x E V. Show that u = v. 4. Let V be a n-dimensional vector space whose basis is {e1,..., en}. Let n x, y E V have the representation r = > Akek, y = 2 akek, where Ak, ak E k=1 k=1 F for all 1 < k = >Aeak k=1 defines an inner product on V. Use this inner product to deduce that every finite dimensional vector space is a Hilbert space. 3. Let (V, ) be an F inner product space. Let u, v E V such that < x, u >= < x,v > for all x E V. Show that u = v. 4. Let V be a n-dimensional vector space whose basis is {e1,..., en}. Let n x, y E V have the representation r = > Akek, y = 2 akek, where Ak, ak E k=1 k=1 F for all 1 < k = >Aeak k=1 defines an inner product on V. Use this inner product to deduce that every finite dimensional vector space is a Hilbert space.

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