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Sobolev Space

The Sobolev space W^k, p(U) consists of all locally summable functions u: U to R such that for each multi-index alpha with |alpha| <= k, the weak derivative D^alphau exists and belongs to L^p(U).

Cambridge IA Numbers and Sets

A proof is a finite sequence of statements, each of which is either an axiom, a previously established result, or follows from earlier statements in the sequence by a valid logical rule, such that the

Discrete Mathematics 1

Continuity (Real Analysis)

Consider the polynomial p(x) = x^3 - 2. Since p(1) = -1 0, the function changes sign on [1, 2], so there must be a root somewhere in between — namely sqrt[3]2. This reasoning is the Intermediate Value

Analysis 1

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Hahn-Banach Theorem

Analysis

In functional analysis, a recurring task is to construct bounded linear functionals with prescribed behaviour. You may need a functional that attains a certain value on a specific vector. You may need one that vanishes on a given subspace but not on a particular element outside it. You may need to c

Updated Apr 18

Uniform Continuity

Analysis

Continuity is a local condition: it asks that $f(x)$ be close to $f(a)$ whenever $x$ is close to $a$, but the meaning of "close" --- the size of $\delta$ --- is permitted to depend on the point $a$. For many purposes in analysis, this freedom is harmless. But for others --- extending a function from

Updated Apr 18

Bounded Linear Operator

Analysis

In finite-dimensional linear algebra, every linear map between normed spaces is automatically continuous. This is a consequence of the equivalence of all norms on $\mathbb{R}^n$: if $T: \mathbb{R}^n \to \mathbb{R}^m$ is linear, then $T$ is represented by a m

Updated Apr 18

Quotient Topology

Analysis

Many of the most natural constructions in geometry and topology begin with an identification: glue two endpoints of an interval to form a circle, collapse the boundary of a disk to a single point to obtain a sphere, or identify opposite edges of a square to build a torus. In each case, we start with

Updated Apr 18

Nets and Filters

Analysis

In metric spaces, sequences characterise the topology completely: a set is closed if and only if it contains the limits of all its convergent sequences, a function is continuous if and only if it pr

Updated Apr 18

Subspace Topology

Analysis

Throughout mathematics, we encounter subsets of topological spaces that we need to study as spaces in their own right. The unit circle $S^1 \subset \mathbb{R}^2$, the closed unit ball $\overline{B}(0, 1) \subset \mathbb{R}^n$, the Cantor set $\mathcal{C} \subset [0, 1]$, and the so

Updated Apr 18

Hausdorff Space

Analysis

In any metric space, a convergent sequence has exactly one limit: if $x_k \to x$ and $x_k \to y$ with $x \neq y$, then for $r = d(x,y)/2 > 0$, the open balls $B(x,r)$ and $B(y,r)$ are disjoint, and the sequence cannot eventually belong to both. This argument uses nothing abou

Updated Apr 18

Measurable Functions

The is built on a deceptively simple idea: to integrate a f, partition its range rather than its domain, and measure the size of each preimage set \{x : f(x) \in I\} for each interval I \subset \mathbb{R}. This strategy succeeds precisely when those pre

Updated Apr 18

Interior

Given a subset A of a X, which points of A are "safely inside" — surrounded by other points of A in every direction — and which sit precariously on the edge? This question is not merely taxonomic. Across analysis, the distinction between interior points and boundary points governs w

Updated Apr 18

Closure

Given a subset A of a X, there are two fundamental questions one can ask about the relationship between A and the ambient space. The first — which points definitely belong to A? — leads to the. The second — which points are "infinitely close" to A, in the sense th

Updated Apr 18

Connectedness

One of the most basic geometric intuitions is that a space can consist of "one piece" or "several pieces." A circle is one piece; a pair of disjoint circles is two pieces. The challenge for topology is to capture this distinction using only the language of open sets — without reference to distance,

Updated Apr 18

Complete Metric Space

Many constructions in analysis proceed by approximation: we build a sequence of objects that get progressively closer to a desired target and then argue that the sequence converges to the target. The gives a purely intrinsic way to detect that a sequence "wants to converge"

Updated Apr 18

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Hahn-Banach Theorem

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Hausdorff Space

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Interior

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Explore Mathematics

Create New Bundle

Browse by Area

Trending This Week

Sobolev Space

Cambridge IA Numbers and Sets

Continuity (Real Analysis)

Recently Updated

All Pages

Hahn-Banach Theorem

Uniform Continuity

Bounded Linear Operator

Quotient Topology

Nets and Filters

Subspace Topology

Hausdorff Space

Measurable Functions

Interior

Closure

Connectedness

Complete Metric Space