Show that Theorem 3.8-1 defines an isometric bijection T: H H', z f = (, z)...

   

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Show that Theorem 3.8-1 defines an isometric bijection T: H H', z f = (, z) which is not linear but conjugate linear, that is, az + BuH āf. + Bf. 3.8 Representation of Functionals on Hilbert Spaces It is of practical importance to know the general form of bounded linear functionals on various spaces. This was pointed out and ex- plained in Sec. 2.10. For general Banach spaces such formulas and their derivation can sometimes be complicated. However, for a Hilbert space the situation is surprisingly simple: 3.8-1 Riesz's Theorem (Functionals on Hilbert spaces). Every bounded linear functional f on a Hilbert space H can be represented in terms of the inner product, namely, (1) f(x) = (x, z) where z depends on f, is uniquely determined by f and has norm (2) ||2| = |- Show that Theorem 3.8-1 defines an isometric bijection T: H H', z f = (, z) which is not linear but conjugate linear, that is, az + BuH āf. + Bf. 3.8 Representation of Functionals on Hilbert Spaces It is of practical importance to know the general form of bounded linear functionals on various spaces. This was pointed out and ex- plained in Sec. 2.10. For general Banach spaces such formulas and their derivation can sometimes be complicated. However, for a Hilbert space the situation is surprisingly simple: 3.8-1 Riesz's Theorem (Functionals on Hilbert spaces). Every bounded linear functional f on a Hilbert space H can be represented in terms of the inner product, namely, (1) f(x) = (x, z) where z depends on f, is uniquely determined by f and has norm (2) ||2| = |-

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