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If $T: X \to Y$ is a linear operator between Banach spaces and its graph is closed in $X \times Y$, then $T$ is continuous.

Every bounded linear functional on a Hilbert space $H$ can be represented as an inner product with a fixed element in $H$.

Every Boolean algebra is isomorphic to a field of sets.

The closed unit ball of the dual space $X^*$ of a normed space $X$ is compact in the weak* topology.

If a family of bounded linear operators from a Banach space $X$ to a normed space $Y$ is pointwise bounded, then it is uniformly bounded on every bounded subset of $X$.

A subset of $C(K)$ (continuous functions on a compact space $K$) is relatively compact if and only if it is bounded and equicontinuous.

In a locally convex space, every compact convex set is the closed convex hull of its extreme points.

Given a Hilbert space $H$, if $a: H \times H \to \mathbb{R}$ is a continuous and coercive bilinear form, and $f: H \to \mathbb{R}$ is a continuous linear functional, then there exists a unique element $u \in H$ such that $a(u, v) = f(v)$ for all $v \in H$.

Every commutative unital C*-algebra is isometrically *-isomorphic to a C*-algebra of continuous functions on some compact Hausdorff space.

In a locally compact Hausdorff space $X$, there is a one-to-one correspondence between Radon measures and positive linear functionals on $C_c(X)$, the space of continuous compactly supported functions on $X$.

If $p: X \to \mathbb{R}$ is a sublinear function and $f: Y \to \mathbb{R}$ is a linear functional on a subspace $Y$ of a vector space $X$ that satisfies $f(y) \leq p(y)$ for all $y \in Y$, then there exists an extension $\overline{f}: X \to \mathbb{R}$ of $f$ to the whole space $X$ such that $\overline{f}(x) \leq p(x)$ for all $x \in X$.

Every contraction mapping on a complete metric space has exactly one fixed point.

Every continuous linear operator from $\mathcal{D}(\Omega)$ to $\mathcal{D}'(\Omega)$ can be represented by a distribution in $\mathcal{D}'(\Omega \times \Omega)$.

If $T: X \to Y$ is a surjective continuous linear operator between Banach spaces $X$ and $Y$, then $T$ is an open map.