1. Let (X, x) and (Y, y) be inner product spaces over the same field F....

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1. Let (X, <, >x) and (Y, <, >y) be inner product spaces over the same field F. Prove that: (a) The map <, >: (X x Y)2 → F, defined by < (x, y), (x', y') >=< x,x' >x + < y;Y' >y is an F inner product on X x Y. (b) < u+u', x+r' >x - < u-u', x– a' >x= 2 < u, x' >x +2 < u', x >x for all x, x', u, u' e X. (с) 4 < и, 2' >-< и + u',х+>x — <и-и',х — " >х +i < u+iu', x+ix' >x -i < u – iu', x– ix' >x for all a, x', u, u' E X if F = C. 2. Let (V, <, >) be an F inner product space. Suppose (an)N and (yn)N are convergent sequences in V to x and y, respectively. Show that: (a) (< xn, Yn >)N is convergent in F to <x, y >. (i.e lim < xn, Yn >=< x, y>=< lim n, lim yn >, whenever lim an = x and lim yn =y). n00 n-00 n-00 n00 (6) lim < axn, z >= a < x, z > for all z E V and for all a E F. n00 3. Let (V, <, >) be an F inner product space. Let u, v E V such that < x, u >=< x, v > for all x € V. Show that u = v. 4. Let V be a n-dimensional vector space whose basis is {e1,.., en}. Let x, y E V have the representation x = E Akek, y = arek, where k, ak € k=1 k=1 F for all 1 < k<n. Show that < x, y >= > Akāk k=1 defines an inner product on V. Use this inner product to deduce that every finite dimensional vector space is a Hilbert space. 1. Let (X, <, >x) and (Y, <, >y) be inner product spaces over the same field F. Prove that: (a) The map <, >: (X x Y)2 → F, defined by < (x, y), (x', y') >=< x,x' >x + < y;Y' >y is an F inner product on X x Y. (b) < u+u', x+r' >x - < u-u', x– a' >x= 2 < u, x' >x +2 < u', x >x for all x, x', u, u' e X. (с) 4 < и, 2' >-< и + u',х+>x — <и-и',х — " >х +i < u+iu', x+ix' >x -i < u – iu', x– ix' >x for all a, x', u, u' E X if F = C. 2. Let (V, <, >) be an F inner product space. Suppose (an)N and (yn)N are convergent sequences in V to x and y, respectively. Show that: (a) (< xn, Yn >)N is convergent in F to <x, y >. (i.e lim < xn, Yn >=< x, y>=< lim n, lim yn >, whenever lim an = x and lim yn =y). n00 n-00 n-00 n00 (6) lim < axn, z >= a < x, z > for all z E V and for all a E F. n00 3. Let (V, <, >) be an F inner product space. Let u, v E V such that < x, u >=< x, v > for all x € V. Show that u = v. 4. Let V be a n-dimensional vector space whose basis is {e1,.., en}. Let x, y E V have the representation x = E Akek, y = arek, where k, ak € k=1 k=1 F for all 1 < k<n. Show that < x, y >= > Akāk k=1 defines an inner product on V. Use this inner product to deduce that every finite dimensional vector space is a Hilbert space. 1. Let (X, <, >x) and (Y, <, >y) be inner product spaces over the same field F. Prove that: (a) The map <, >: (X x Y)2 → F, defined by < (x, y), (x', y') >=< x,x' >x + < y;Y' >y is an F inner product on X x Y. (b) < u+u', x+r' >x - < u-u', x– a' >x= 2 < u, x' >x +2 < u', x >x for all x, x', u, u' e X. (с) 4 < и, 2' >-< и + u',х+>x — <и-и',х — " >х +i < u+iu', x+ix' >x -i < u – iu', x– ix' >x for all a, x', u, u' E X if F = C. 2. Let (V, <, >) be an F inner product space. Suppose (an)N and (yn)N are convergent sequences in V to x and y, respectively. Show that: (a) (< xn, Yn >)N is convergent in F to <x, y >. (i.e lim < xn, Yn >=< x, y>=< lim n, lim yn >, whenever lim an = x and lim yn =y). n00 n-00 n-00 n00 (6) lim < axn, z >= a < x, z > for all z E V and for all a E F. n00 3. Let (V, <, >) be an F inner product space. Let u, v E V such that < x, u >=< x, v > for all x € V. Show that u = v. 4. Let V be a n-dimensional vector space whose basis is {e1,.., en}. Let x, y E V have the representation x = E Akek, y = arek, where k, ak € k=1 k=1 F for all 1 < k<n. Show that < x, y >= > Akāk k=1 defines an inner product on V. Use this inner product to deduce that every finite dimensional vector space is a Hilbert space. 1. Let (X, <, >x) and (Y, <, >y) be inner product spaces over the same field F. Prove that: (a) The map <, >: (X x Y)2 → F, defined by < (x, y), (x', y') >=< x,x' >x + < y;Y' >y is an F inner product on X x Y. (b) < u+u', x+r' >x - < u-u', x– a' >x= 2 < u, x' >x +2 < u', x >x for all x, x', u, u' e X. (с) 4 < и, 2' >-< и + u',х+>x — <и-и',х — " >х +i < u+iu', x+ix' >x -i < u – iu', x– ix' >x for all a, x', u, u' E X if F = C. 2. Let (V, <, >) be an F inner product space. Suppose (an)N and (yn)N are convergent sequences in V to x and y, respectively. Show that: (a) (< xn, Yn >)N is convergent in F to <x, y >. (i.e lim < xn, Yn >=< x, y>=< lim n, lim yn >, whenever lim an = x and lim yn =y). n00 n-00 n-00 n00 (6) lim < axn, z >= a < x, z > for all z E V and for all a E F. n00 3. Let (V, <, >) be an F inner product space. Let u, v E V such that < x, u >=< x, v > for all x € V. Show that u = v. 4. Let V be a n-dimensional vector space whose basis is {e1,.., en}. Let x, y E V have the representation x = E Akek, y = arek, where k, ak € k=1 k=1 F for all 1 < k<n. Show that < x, y >= > Akāk 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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