10.2 Reflexive spaces and their properties

Reflexive spaces are a crucial concept in functional analysis, linking a space to its dual and bidual. They include finite-dimensional normed spaces, Hilbert spaces, and certain sequence and Lebesgue spaces, but not all Banach spaces are reflexive.

Understanding reflexivity helps analyze properties of direct sums and products of spaces. It's a stronger condition than completeness, offering deeper insights into the structure of normed spaces and their relationships to their dual spaces.

Reflexive Spaces

Examples of reflexive spaces

• Finite-dimensional normed spaces are always reflexive due to the natural isomorphism between a finite-dimensional space and its double dual
• Hilbert spaces, such as the space of square-integrable functions $L^2(\Omega)$ over a measure space $\Omega$, are reflexive because the Riesz representation theorem provides a natural isometric isomorphism between the space and its dual
• The sequence spaces $\ell^p$ for $1 < p < \infty$ are reflexive, as the dual of $\ell^p$ is isometrically isomorphic to $\ell^q$, where $\frac{1}{p} + \frac{1}{q} = 1$
• The Lebesgue spaces $L^p(\Omega)$ for $1 < p < \infty$ and a measure space $\Omega$ are reflexive, with the dual of $L^p(\Omega)$ being isometrically isomorphic to $L^q(\Omega)$, where $\frac{1}{p} + \frac{1}{q} = 1$
• The sequence spaces $\ell^1$ and $\ell^\infty$ are not reflexive, as their duals are $\ell^\infty$ and $(\ell^1)^{**}$, respectively, which are not isometrically isomorphic to the original spaces
• The Lebesgue spaces $L^1(\Omega)$ and $L^\infty(\Omega)$ for a measure space $\Omega$ are not reflexive, as their duals are $L^\infty(\Omega)$ and the space of finitely additive measures on $\Omega$, respectively, which are not isometrically isomorphic to the original spaces

Reflexivity of closed subspaces

• The dual space $Y^*$ can be identified with the quotient space $X^*/Y^\perp$, where $Y^\perp = \{f \in X^* : f(y) = 0 \text{ for all } y \in Y\}$ is the annihilator of $Y$
• The bidual $Y^{**}$ can be identified with a subspace of $X^{**}$ via the composition of natural embeddings $Y \hookrightarrow X \hookrightarrow X^{**}$, where the first embedding is the inclusion map and the second is the natural embedding $J_X$
• The restriction of the surjective natural embedding $J_X: X \to X^{**}$ to $Y$, denoted by $J_Y: Y \to Y^{**}$, is also surjective because $Y^{**}$ is a subspace of $X^{**}$ and $J_X$ maps $Y$ onto this subspace
• The surjectivity of $J_Y$ implies that $Y$ is reflexive, as the natural embedding from $Y$ to its bidual is surjective

Properties of Reflexive Spaces

Reflexivity in sums and products

• Direct sum:
  1. The dual space $(X \oplus Y)^*$ is isometrically isomorphic to $X^* \oplus Y^*$ by the definition of the dual of a direct sum
  2. The bidual space $(X \oplus Y)^{**}$ is isometrically isomorphic to $X^{**} \oplus Y^{**}$ by the definition of the bidual of a direct sum
  3. The surjectivity of the natural embeddings $J_X: X \to X^{**}$ and $J_Y: Y \to Y^{**}$ implies the surjectivity of their direct sum $J_{X \oplus Y}: X \oplus Y \to (X \oplus Y)^{**}$
  4. The surjectivity of $J_{X \oplus Y}$ proves that $X \oplus Y$ is reflexive
• Product:
  1. The dual space $(\prod_{i \in I} X_i)^*$ is isometrically isomorphic to the direct sum $\bigoplus_{i \in I} X_i^*$ by the definition of the dual of a product space
  2. The bidual space $(\prod_{i \in I} X_i)^{**}$ is isometrically isomorphic to the product $\prod_{i \in I} X_i^{**}$ by the definition of the bidual of a product space
  3. The surjectivity of each natural embedding $J_{X_i}: X_i \to X_i^{**}$ implies the surjectivity of their product $J_{\prod X_i}: \prod_{i \in I} X_i \to (\prod_{i \in I} X_i)^{**}$
  4. The surjectivity of $J_{\prod X_i}$ proves that $\prod_{i \in I} X_i$ is reflexive

Reflexivity vs completeness

• The Banach-Steinhaus theorem states that the dual of a normed space is always a Banach space (complete normed space)
• A reflexive space $X$ is isometrically isomorphic to its bidual $X^{**}$, which is a Banach space, implying that $X$ itself is a Banach space
• The sequence spaces $\ell^1$, $\ell^\infty$ and the Lebesgue spaces $L^1(\Omega)$, $L^\infty(\Omega)$ for a measure space $\Omega$ are examples of Banach spaces that are not reflexive
• Reflexivity is a stronger property than completeness for normed spaces, as it requires the natural embedding from the space to its bidual to be surjective, while completeness only requires Cauchy sequences to converge within the space