10.1 Bidual spaces and natural embeddings

Bidual spaces take us on a journey through layers of linear functionals. We start with a normed space, create its dual of continuous linear functionals, then form the bidual of functionals on functionals. This process reveals deep connections between spaces.

Reflexivity is a key concept in this exploration. A space is reflexive if it's isomorphic to its bidual, meaning they're essentially the same. Non-reflexive spaces, however, have biduals that are strictly larger, showcasing the complexity of functional analysis.

Bidual Spaces

Construction of bidual spaces

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• Start with a normed linear space $X$ (Banach spaces, Hilbert spaces)
• Form the dual space $X^*$ consisting of all continuous linear functionals on $X$
  • Continuous linear functionals map elements of $X$ to scalars while preserving linearity and continuity
• Construct the bidual space $X^{**}$ as the dual space of $X^*$
  • Elements of $X^{**}$ are continuous linear functionals on $X^*$, assigning scalars to functionals
• $X^{**}$ is called the "dual of the dual" or "functionals on functionals"
  • Functionals in $X^{**}$ take functionals from $X^*$ as input and output scalar values

Natural embedding and bidual relations

• Define the natural embedding $J: X \to X^{**}$ as $J(x)(f) = f(x)$ for $x \in X$ and $f \in X^*$
  • $J$ maps elements of $X$ to elements of $X^{**}$
• $J$ is a linear map preserving the norm: $\|J(x)\| = \|x\|$ for all $x \in X$
• $J$ is always injective (one-to-one), embedding $X$ into $X^{**}$
  • Allows viewing $X$ as a subspace of its bidual $X^{**}$
• Injectivity of $J$ means distinct elements of $X$ map to distinct elements of $X^{**}$
  • Ensures no information is lost when mapping from $X$ to $X^{**}$

Reflexive Spaces

Isomorphisms in reflexive spaces

• A space $X$ is reflexive if the natural embedding $J: X \to X^{**}$ is surjective (onto)
  • Every element of $X^{**}$ is the image of some element in $X$ under $J$
• In reflexive spaces, $J$ is an isometric isomorphism between $X$ and $X^{**}$
  • $J$ is bijective (one-to-one and onto) and preserves the norm
• Proof of isometric isomorphism in reflexive spaces:
  1. $J$ is surjective by the definition of reflexivity
  2. $J$ is injective and norm-preserving for any space
  3. Combining surjectivity, injectivity, and norm-preservation, $J$ is an isometric isomorphism
• Reflexive spaces can be identified with their biduals via the isometric isomorphism $J$
  • $X$ and $X^{**}$ are essentially the same space in the reflexive case

Non-reflexive spaces vs biduals

• Examples of non-reflexive spaces:
  1. $c_0$: space of sequences converging to zero
     • Its bidual is $\ell^{\infty}$, the space of bounded sequences
  2. $L^1(\mathbb{R})$: space of absolutely integrable functions on the real line
     • Its bidual is $L^{\infty}(\mathbb{R})$, the space of essentially bounded functions
• In non-reflexive spaces, the bidual is strictly larger than the original space
  • Natural embedding $J$ is injective but not surjective
• Non-reflexive spaces have elements in the bidual that do not correspond to any element in the original space
  • Demonstrates that not all spaces are isomorphic to their biduals
• Non-reflexivity highlights the distinction between a space and its bidual
  • Bidual may contain additional elements not present in the original space