10.3 Riesz representation theorem

The Riesz representation theorem is a game-changer in Hilbert spaces. It shows that every bounded linear functional can be represented as an inner product with a unique vector. This connection between abstract functionals and concrete vectors is super useful.

This theorem creates a perfect match between a Hilbert space and its dual space. It's like finding your space's mirror image, where every functional has a corresponding vector. This idea pops up all over the place in math and physics.

Linear Functionals and Dual Space

Definition and Properties of Linear Functionals

Top images from around the web for Definition and Properties of Linear Functionals

• 

  Graphs of Linear Functions · Precalculus View original

  Is this image relevant?

• 

  Dual space (linear algebra) - Knowino View original

  Is this image relevant?

• 

  Dual space (linear algebra) - Knowino View original

  Is this image relevant?

• 

  Graphs of Linear Functions · Precalculus View original

  Is this image relevant?

• 

  Dual space (linear algebra) - Knowino View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and Properties of Linear Functionals

• 

  Graphs of Linear Functions · Precalculus View original

  Is this image relevant?

• 

  Dual space (linear algebra) - Knowino View original

  Is this image relevant?

• 

  Dual space (linear algebra) - Knowino View original

  Is this image relevant?

• 

  Graphs of Linear Functions · Precalculus View original

  Is this image relevant?

• 

  Dual space (linear algebra) - Knowino View original

  Is this image relevant?

1 of 3

• Linear functional is a linear map from a vector space $V$ to its underlying field $\mathbb{F}$ (real numbers or complex numbers)
• Satisfies linearity properties for all vectors $x, y \in V$ and scalars $a \in \mathbb{F}$:
  • Additivity: $f(x + y) = f(x) + f(y)$
  • Homogeneity: $f(ax) = af(x)$
• Examples of linear functionals:
  • Evaluation functional: $f(p) = p(a)$ for a fixed $a \in \mathbb{R}$ and polynomials $p$
  • Integration functional: $f(g) = \int_a^b g(x) dx$ for a fixed interval $[a, b]$ and functions $g$

Bounded Linear Functionals and Norm

• Bounded linear functional has a finite operator norm, defined as:
  • $\|f\| = \sup\{|f(x)| : \|x\| \leq 1\}$
  • Measures the maximum absolute value of $f$ on the unit ball of $V$
• Bounded linear functionals form a normed vector space, denoted as $V^*$ or $B(V, \mathbb{F})$
• Examples of bounded linear functionals:
  • Evaluation functional on the space of continuous functions $C([a, b])$
  • Integration functional on the space of integrable functions $L^1([a, b])$

Dual Space and its Properties

• Dual space $V^*$ is the vector space of all bounded linear functionals on $V$
• Dual space is a Banach space (complete normed vector space) when $V$ is a normed vector space
• Dual space of a Hilbert space $H$ is isometrically isomorphic to $H$ itself (Riesz representation theorem)
• Examples of dual spaces:
  • Dual of $\ell^p$ space is isometrically isomorphic to $\ell^q$ space, where $1/p + 1/q = 1$
  • Dual of $L^p([a, b])$ space is isometrically isomorphic to $L^q([a, b])$ space, where $1/p + 1/q = 1$

Riesz Representation Theorem

Statement and Significance of the Theorem

• Riesz representation theorem states that for every bounded linear functional $f$ on a Hilbert space $H$, there exists a unique vector $y \in H$ such that:
  • $f(x) = \langle x, y \rangle$ for all $x \in H$
  • $\|f\| = \|y\|$
• Establishes a one-to-one correspondence between the Hilbert space $H$ and its dual space $H^*$
• Allows the representation of abstract linear functionals as inner products with concrete vectors
• Fundamental result in functional analysis with applications in quantum mechanics, signal processing, and other fields

Isomorphism between Hilbert Space and its Dual

• Riesz representation theorem induces an isometric isomorphism between $H$ and $H^*$
• Isomorphism is a bijective linear map that preserves the vector space structure
• Isometric isomorphism additionally preserves the norm, i.e., $\|f\| = \|y\|$ for the corresponding $f \in H^*$ and $y \in H$
• Examples of isometric isomorphisms:
  • $\ell^2$ space is isometrically isomorphic to its dual $(\ell^2)^*$
  • $L^2([a, b])$ space is isometrically isomorphic to its dual $(L^2([a, b]))^*$

Reflexive Spaces and their Characterization

• Reflexive space is a Banach space $V$ such that the canonical embedding $J: V \to V^{**}$ is surjective
• Canonical embedding $J$ maps each vector $x \in V$ to the evaluation functional $J(x) \in V^{**}$ defined by:
  • $J(x)(f) = f(x)$ for all $f \in V^*$
• Hilbert spaces are reflexive, as a consequence of the Riesz representation theorem
• Examples of reflexive spaces:
  • $L^p([a, b])$ spaces for $1 < p < \infty$
  • $\ell^p$ spaces for $1 < p < \infty$