## Linear Operators: Spectral theory |

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Page 1179

which sends a scalar - valued function with the Fourier transform f ( 5 ) into the

vector - valued function whose nth component has the Fourier transform ( 5 )

defined ...

**PROOF**. We saw in the course of proving Theorem 25 that the mapping M Kwhich sends a scalar - valued function with the Fourier transform f ( 5 ) into the

vector - valued function whose nth component has the Fourier transform ( 5 )

defined ...

Page 1724

) = ( 1 , Sg ) forf in D ( T ) and g in D ( S ) . By Green's formula , proved in the last

paragraph of Section 2 , this equation is valid if | and g are in C9 * ( I ) . It follows ...

**Proof**. By the preceding lemma and by Corollary 11 it suffices to show that ( Tj , g) = ( 1 , Sg ) forf in D ( T ) and g in D ( S ) . By Green's formula , proved in the last

paragraph of Section 2 , this equation is valid if | and g are in C9 * ( I ) . It follows ...

Page 1750

We shall see , however , that this fact is needed in the course of the

Theorem 1 , and shall prove it by a direct method where it is needed . Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient ...

We shall see , however , that this fact is needed in the course of the

**proof**ofTheorem 1 , and shall prove it by a direct method where it is needed . Remark 2.

The theorem is false if no boundedness restriction is imposed on the coefficient ...

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Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

Linear Operators, Part 2: Spectral Theory, Self Adjoint Operators in Hilbert ... Nelson Dunford,Jacob T. Schwartz No preview available - 1988 |

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