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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.

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 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 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 $\mathbb{R}$.

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 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 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.

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.

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 Hilbert space is a complete normed vector space with an inner product that induces the norm. Example: The space $L^2([0, 1])$ of square-integrable functions is a Hilbert space.

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.

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.

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 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 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 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 $\mathbb{R}$.

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.