(每題10分)
1.	Show that , n= 1, 2, 3,…, implies that = 0.
2.	Show that a linear operator in a Banach space is continuous if and only if its null space is closed.
3.	Show that en = ( 0, … 0, 1, 0, … ), where 1 is in the n-th place, converges weakly to (0,0,0,...) in l2.
4.	Show that a finite dimensional subspace of a normed vector space is closed.
5.	Let X be a separable Banach space and A be a weakly compact subset of X . Show that the weak topology on A is a metric topology.
6.	Show that a finite-dimensional normed linear space is reflexive.
7.	Let be the space of functions that are defined and continuous on the unit disk , and analytic on |z|<1. Define a norm for by . Show that is separable.
8.	Show that a self-adjoint operator on a Hilbert space is symmetric.
9.	Let X be a separable Hilbert space and {en} be an orthonormal basis of X , and T be a compact operator of X into a normed linear space. Show that Ten → 0 as n → .
10.	Let X be the space of real sequences with and . Show that the closed unit ball in X has no extremal points.

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