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May 25th, 2014, 02:18 AM   #1
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Direct sum of Banach spaces

Theorem. Let F = ($X_j$, $j$ elem of J) be a family of Banach spaces over field $R$ or $C$ and let 1<=p<$\infty$. With X we note the set of all functions x from the index set J into the union of sets $X_j$ with the following properties:

1. x(j) elem of $X_j$, $j$ elem of J
2. sum j € J of p-th powers of norms of vectors x(j) < &\infty$.

If we equip X with the usual structure of function vector space, then X is a Banach space relative to the norm

(1/p)-th power of the sum in 2.

X is called direct sum of family ($X_j$, $j$ elem of J) of Banach spaces and is noted $l_p$(F).

I wonder what the index set is, since we sum reals or complex numbers.
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May 25th, 2014, 02:26 AM   #2
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2. sum j € J of p-th powers of norms of vectors x(j) < infinity.

If we equip X with the usual structure of function vector space, then X is a Banach space relative to the norm (1/p)-th power of the sum in 2.

X is called direct sum of the family.

I wonder what the index set is, because we sum real or complex numbers.
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May 25th, 2014, 08:09 AM   #3
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I guess J is N or a finite set.

Proof. We first have to show X is a vector space. For x, y elem X, z = x + y and $\lambda$x are defined as the sum and product of function and $\lambda$.

Obviously x elem X implies $\lambda$x elem X (because $X_j$ is a vector space)
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