Any non zero bounded linear functional on a Banach space is an open map.

Let A be an operator on a Hilbert space .H Show if Ax A x ∗ = for every ,H x ∈ then A is normal. Is it converse true? Justify

Check whether the identity map on an infinite dimensional space is compact.

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 ,.

The space l³ is a Hilbert space

Are Hahn-Banach extensions always unique? Justify.

State the principle of uniform boundedness. Use it to show that a set E in a normed space X is bounded if )E(f is bounded in K for every f ∈ X′.

In a Hilbert space. Prove that x _n → x provided