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daishin
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Homework Statement
Let 1[tex]\leq[/tex]p[tex]\leq[/tex][tex]\infty[/tex] and suppose ([tex]\alpha_{ij}[/tex] is a matrix such that (Af)(i)=[tex]\sum^{\infty}_{j=1}[/tex][tex]\alpha[/tex][tex]_{ij}[/tex]f(j) defines an element Af of [tex]l^{p}[/tex] for every f in [tex]l^{p}[/tex]. Show that A is a bounded linear functional on [tex]l^{p}[/tex]
Homework Equations
Isn't this obvious if we apply theorem that says following are equivalent for A:X-->X a linear transformation on normed space?
(a)A is bounded linear functional
(b)A is continuous at some point
(c)There is a positive constant c such that ||Ax||[tex]\leq[/tex]c||x|| for all x in X.
The Attempt at a Solution
Isn't the contiunuity of f obvious? So by the theorem, I think A is bounded linear functional on [tex]l^{p}[/tex]. Could you guys correct me if I am wrong? Or if I am right could you just say it is right?
Thanks
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