The space l³ is a Hilbert space

If H is a Hilbert space and SCH, show that . S S ⊥ ⊥⊥⊥ = When S is the same as ? S ⊥⊥ Justify.

The image of a Cauchy sequence under a bounded linear map is also a Couchy sequence.

Let X be an inner product space with the inner product given by <, > . For ,X x ∈ define the function given by the non negative square root of < x, x > . Show that defines a norm on X and for all .X ,x y∈ Also show that for all ,X ,.

Let ]1,0[C X = with Sup norm defined by f = Sup{| )x(f |}. ]1

Let T be a linear map defined on X by

Show that T is a bounded linear map such that T = .1

Every bounded linear map on a complex Banach space has an eigen value.

Let X be a vector space. Let be two norms on .X When are these norms said to be equivalent? Justify your answer.

Let . X  = For  x=(x_1,x₂,x₃)

Let

Show that and are equivalent.