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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 132.34902

**Autor: ** Erdös, Pál; Shapiro, H.S.; Shields, A.L.

**Title: ** Large and small subspaces of Hilbert space (In English)

**Source: ** Mich. Math. J. 12, 169-178 (1965).

**Review: ** This paper is concerned with the properties of closed subspaces V of the sequential Hilbert space l_{2} and of L_{2}(0,1). We shall suffice by quoting the following interesting results of this paper which speak for themselves.

Theorem 1. Let V be a closed linear subspace of l_{2}, and let **{**\rho_{n}**}** be given with \rho_{n} \geq 0 and **sum** \rho_{n}^{2} < oo. If |x(n)| = O(\rho_{n}) for all x in V, then V is finite-dimensional.

Theorem 3. If V is a closed subspace of l_{2} and V \subset I_{p} for some 1 \leq p < 2, then V is finite-dimensional.

Theorem 4. If \rho_{n} \geq 0 and **sum** \rho_{n}^{2} = oo, then there exists an infinite-dimensional subspace V of I_{2} such that **sum**|x(n)| \rho(n) = oo for all x \ne 0 in V. In the case of L_{2} (0,1) the situation is different.

The authors quote the well-known result from the theory of Fourier series that there exists an infinite-dimensional closed subspace V of L_{2} (0,1) such that V \subset L_{q} for all 1 \leq q < oo and in fact satisfies the condition that **int** \exp**{**c|f(x)|^{2}**}**dx < oo for all c > 0 and all f in V. Then it is shown that if \phi is convex, continuous and strictly increasing on [0,oo) with \phi(0) = 0 and \phi(x)e^{-cx2} ––> oo as x ––> oo for all c > 0, then **int** \phi(|f|) < oo for all f in V implies that V is finite dimensional. Let V be a closed linear subspace of I_{2}. Then there exist elements \lambda_{n} (n = 1,2,...) in V such that (x,\lambda_{n}) = x(n) for all x in V and dim V = **sum** ||\lambda_{n}||^{2}. This result is used to prove the following theorem. Theorem 9. Let \phi(z) = **sum** a_{n} z^{n} be an inner function. Then **sum** n|a_{n}|^{2} = dim(\phi H_{2})\bot. Thus the Dirichlet integral of \phi is finite (and is then an integral multiple of \pi) if and only if \phi is a finite Blaschke product. The paper finishes with the following question: Does H_{2} contain an infinite dimensional closed subspace. V with |f(z)| = O(1/(1-|z|)^{1/4}) (|z| < 1).

**Reviewer: ** W.A.J.Luxemburg

**Classif.: ** * 46C05 Geometry and topology of inner product spaces

**Index Words: ** functional analysis

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