State whether the following statements True or False ? Justify your answers :
5\times 2=10
(a)
||
.
||definedonR^(n)
as :
||x||=\sum_(j=1)^n |a_(j)| for n=(a_(1),a_(2),dotsdots,a_(n))inR^(n)
is a norm. (b)
C_(0)
is a Banach space. (c) If A is the right shift operator on
l^(2)
, then the eigen spectrum is non-empty. (d) If a normed space is reflexive, then so is its dual space. (e) If a normed linear space
x
is finite dimesnional, then so is
x^(')
.

Question

State whether the following statements True or False ? Justify your answers :

5\times 2=10

(a)

||

.

||definedonR^(n)

as :

||x||=\sum_(j=1)^n |a_(j)| for n=(a_(1),a_(2),dotsdots,a_(n))inR^(n)

is a norm. (b)

C_(0)

is a Banach space. (c) If A is the right shift operator on

l^(2)

, then the eigen spectrum is non-empty. (d) If a normed space is reflexive, then so is its dual space. (e) If a normed linear space

x

is finite dimesnional, then so is

x^(')

.

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