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A norm is a function that assigns a strictly positive length or size to all vectors in a vector space, except for the zero vector. Example: Euclidean norm on $\mathbb{R}^2$, $\|x\|_2 = \sqrt{x_1^2 + x_2^2}$.

A normed space is a vector space on which a norm is defined. Example: $(\mathbb{R}^n, \|\cdot\|_2)$ is a normed space with the Euclidean norm.

The unit ball in a normed space is the set of all points that have a norm less than or equal to 1. Example: In $\mathbb{R}^2$ with the Euclidean norm, the unit ball is the set of all points within and on the circle of radius 1.

An open ball is the set of all points in a normed space whose distance from a center point is less than a given radius. Example: In $\mathbb{R}^2$, the open ball centered at the origin with radius 1 is the set of all points strictly inside the circle of radius 1.

A closed ball in a normed space includes all points within a certain radius of a center point, as well as the points on the boundary. Example: In $\mathbb{R}^2$ with the Euclidean norm, the closed ball of radius 1 includes all the points within and on the circle of radius 1 around the origin.

A subset of a normed space is bounded if there exists a real number that is greater than or equal to the norm of all elements in the subset. Example: The set $\{(x, y) \in \mathbb{R}^2 | x^2 + y^2 \leq 9\}$ is bounded in the Euclidean space $\mathbb{R}^2$.

A sequence in a normed space converges if the norm of the difference between the sequence elements and some limit point approaches zero as the sequence progresses. Example: The sequence $\{1/n\}$ in $\mathbb{R}$ converges to 0.

A Cauchy sequence is a sequence where the norms of the differences between sequence elements eventually become arbitrarily small, regardless of how far apart the elements are along the sequence. Example: The sequence $\{1/n\}$ in $\mathbb{R}$ is a Cauchy sequence.

A complete normed space (also known as a Banach space) is one in which every Cauchy sequence converges to a point within the space. Example: The Euclidean space $\mathbb{R}^n$ with the standard norm is complete.

A function between normed spaces is continuous if small changes in the input result in small changes in the output with respect to their respective norms. Example: The function $f(x) = 3x$ is continuous in $\mathbb{R}$.

A linear operator is a linear transformation between two normed spaces. It preserves addition and scalar multiplication. Example: The derivative operator $D(f) = f'$ is a linear operator on the normed space of differentiable functions.

The dual space of a normed space is the set of all continuous linear functionals on that space. Example: For the normed space $\mathbb{R}^n$, the dual space consists of all linear functions from $\mathbb{R}^n$ to $\mathbb{R}$.

An $L^p$ space is a function space defined for $1 \leq p < \infty$ where the $p$-th power of the absolute value of the function is Lebesgue integrable. Example: The space of all square-integrable functions on an interval is an $L^2$ space.

The supremum norm (or infinity norm) is the maximum absolute value of a function over its domain. Example: For $f(x) = \sin x$ on $[0, 2\pi]$, the supremum norm is 1.

The Banach fixed-point theorem states that a contraction mapping on a complete metric space has a single fixed point. Example: The map $f(x) = x/2$ on $[0, 1]$ has a fixed point at $x = 0$.

A separable space is a normed space that contains a countable dense subset. Example: The space $\mathbb{R}$ with the standard norm is separable because the rational numbers form a countable dense subset.

A continuous linear functional on a normed space is a linear functional that is also continuous. Example: The functional $f(x) = x(0)$ on the space of continuous functions on $[0, 1]$ is continuous.

A bilinear map is a function that is linear in each of two arguments separately when the other is held fixed. Example: The dot product on $\mathbb{R}^n$ is a bilinear map.

A subspace of a normed space is a subset that is also a normed space with the induced norm. Example: The set of all functions satisfying $f(0) = 0$ in the space of all continuous functions on $[0, 1]$ is a subspace.