3. Suppose {e_1, e_2, ...} is an orthonormal basis for H and for each n there is a vector Ae_n in H such that ? ||Ae_n|| < ?. Show that A has an unique extension to a bounded operator on H.
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First, we need to show that A can be extended to a linear operator on the whole Hilbert space. To do this, we define the action of A on an arbitrary vector x in the Hilbert space as follows: $$Ax = A(c_1 e_1 + c_2 e_2) = c_1 A e_1 + c_2 A e_2$$ where $c_1$ Show more…
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