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

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6. 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 + Bu+ āf, + Bf.. 7. Show that the dual space H' of a Hilbert space H is a Hilbert space with inner product (, ·), defined by 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 bounded linear functional f on a Hilbert space H can be represented in terms of the inner product, namely, on Hilbert spaces). Every (1) f(x) = (x, z) where z depends on f, is uniquely determined by f and has norm (2) 6. 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 + Bu+ āf, + Bf.. 7. Show that the dual space H' of a Hilbert space H is a Hilbert space with inner product (, ·), defined by 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 bounded linear functional f on a Hilbert space H can be represented in terms of the inner product, namely, on Hilbert spaces). Every (1) f(x) = (x, z) where z depends on f, is uniquely determined by f and has norm (2)

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