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The Hahn-Banach theorem states that if $p$ is a sublinear function and $f$ is a linear functional bounded above by $p$ on a subspace, then there exists an extension of $f$ over the whole space without increasing its norm. Example: Extending a functional defined on a subspace of continuous functions to all continuous functions without increasing the norm of the functional.

A Banach space $X$ is called reflexive if every continuous linear functional on $X$ can be represented as an inner product with a vector in $X$. Example: The $L^p$ spaces for $1 < p < \\infty$ are reflexive.

Weak topology on a Banach space is the coarsest topology such that all continuous linear functionals remain continuous. Example: In the weak topology, a sequence converges if and only if it converges pointwise on the dual space.

Weak-* topology on the dual space $X^*$ is the topology of pointwise convergence on $X$. Example: A sequence $(f_n)$ in the dual space $X^*$ converges to $f$ in the weak-* topology if for all $x$ in $X$, $f_n(x) \\to f(x)$ as $n \\to \\infty$.

The dual space $X^*$ of a normed space $X$ consists of all continuous linear functionals on $X$. Example: The dual space of $L^p([a, b])$ where $1 < p < \\infty$ is $L^q([a, b])$ with $\\frac{1}{p} + \\frac{1}{q} = 1$.

The Banach-Alaoglu theorem states that the closed unit ball in the dual space $X^*$ of a normed space $X$ is compact in the weak-* topology. Example: The set of norm-bounded linear functionals on $\\ell^1$ is compact in the weak-* topology.

The bidual space $X^{**}$ is the dual space of the dual space $X^*$, consisting of all continuous linear functionals on $X^*$. Example: For $\\ell^1$ space, the bidual space is isometrically isomorphic to $\\ell^{\\infty}$.

Given a subset $S$ of a normed space $X$, its annihilator $S^\perp$ is the set of all continuous linear functionals in $X^*$ that vanish on $S$. Example: If $S$ is the subspace of $X$ spanned by a vector $x$, then $S^\perp$ consists of all functionals that have $x$ in their kernel.

The Goldstine theorem asserts that the image of the unit ball of a Banach space $X$ under the canonical embedding into its bidual $X^{**}$ is weak-* dense in the unit ball of $X^{**}$. Example: The sequence space $\\ell^1$ can be densely embedded into $\\ell^\infty$, which is its bidual.

A Schauder basis is a sequence $(x_n)$ in a Banach space such that every element of the space can be uniquely expressed as a convergent series of these basis vectors. Example: The sequence of monomials $(1, x, x^2, x^3, \\dots)$ forms a Schauder basis for the space of continuous functions on $[0, 1]$ with the sup-norm.

The algebraic dual of a vector space $X$ is the set of all linear functionals on $X$, not necessarily continuous. Example: The algebraic dual of a finite-dimensional vector space is also finite-dimensional and has the same dimension as the space.

James' theorem characterizes reflexivity in Banach spaces, stating that a Banach space is reflexive if and only if every continuous linear functional on the space attains its maximum on the closed unit ball. Example: The space $L^p([a, b])$ for $1 < p < \\infty$ is reflexive, so every continuous linear functional attains its supremum on the closed unit ball of $L^p$.

A Banach space is called separable if it contains a countable dense subset. Example: The space of square-summable sequences $\\ell^2$ is separable as the finite sequences with rational coefficients form a countable dense subset.

The double dual of a Banach space $X$ is the bidual $X^{**}$. Example: If $X = \\ell^1$, then its bidual $X^{**}$ is isometrically isomorphic to $\\ell^\infty$.

For $1 < p < \\infty$, the dual space of $L^p$ is $L^q$ where $1/p + 1/q = 1$. Example: The dual space of $L^2(\\mathbb{R})$ is also $L^2(\\mathbb{R})$, exhibiting self-duality.

Conjugate spaces in functional analysis refer to the dual spaces. Example: The conjugate space of $c_0$ (the space of sequences converging to zero) is $\\ell^1$.

The space $c_0$ consists of all sequences converging to zero, and its dual is $\\ell^1$. Example: If a sequence $(a_n) \\in c_0$, then any functional in $\\ell^1$ can be applied by summing $\\sum a_n f(n)$ where $f \\in \\ell^1$.

The Riesz Representation theorem states that for a Hilbert space $H$, every continuous linear functional $f$ on $H$ can be represented as an inner product with some element $y \in H$, i.e., $f(x) = \\langle x, y \\rangle$. Example: In $L^2([0, 1])$, the functional $f(g) = \\int_0^1 g(x)h(x) dx$ for some fixed $h \in L^2([0, 1])$ is such a representation.

A duality pairing is a bilinear form on a product space $X \times X^*$ such that for each non-zero $x \in X$, there exists an $f \in X^*$ with the pairing non-zero, and vice versa. Example: The integral $\\langle f, g \\rangle = \\int_X f(x)g(x) dx$ defines a duality pairing between $L^p$ and $L^q$ spaces where $1 < p, q < \\infty$ and $\\frac{1}{p}+\\frac{1}{q}=1$.