By Lewis G.N.

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**Example text**

In a normed vector space X is said to be completet fi its linear span is dense in X , that is, if for each vector x and each E > 0 there is a finite linear combination c 1x, + ’ . cnx, such that + IIX - (C’XI + . . + c,xn)II < E. It is a direct and important consequence of the Hahn-Banach theorem that the completeness of a sequence of vectors {x,} in X is equivalent to the following condition : if p E X * (the topological dual of X ) and if p ( x , ) = 0 ( n = 1,2,3,. ), then p = 0. When X is a Hilbert space, the Riesz representation theorem shows that all is well-the former definition (see Section 3) agrees with the present one.

D) Show that { f,,} is a Hilbert basis if and only if there exists a constant B > 0 such that for arbitrary scalars c l , . , c, (n = 1 , 2 , 3 , . ). Let { j ; , }be a basis for a Hilbert space H and let {g,} be its biorthogonal basis. } are equivalent if and only if { J n }is a Riesz basis. 6. Let { j i I }be a basis for a Hilbert space H and let g, = A,f,, where 5. Prove or disprove: 9 {fn} and {g,,} are equivalent. The Stability of Bases in Banach Spaces Two mathematical objects that are in some sense “close” to each other often share common properties.

C,xn)II < E. It is a direct and important consequence of the Hahn-Banach theorem that the completeness of a sequence of vectors {x,} in X is equivalent to the following condition : if p E X * (the topological dual of X ) and if p ( x , ) = 0 ( n = 1,2,3,. ), then p = 0. When X is a Hilbert space, the Riesz representation theorem shows that all is well-the former definition (see Section 3) agrees with the present one. Example 1. , p) is a a-finite measure space and 1 5 p < 00, then the Riesz representation theorem shows that the dual of Lp(p)can be identified with Lq(p), where l / p l / q = 1 .

### A New Equation for the Distribution of Radiant Energy by Lewis G.N.

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