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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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Measure Space

Analysis

How do we assign "size" to a set? For finite collections of points, counting works. For intervals on the real line, length works. For rectangles in the plane, area works. But modern analysis demands far more: we must assign sizes to highly irregular sets — the preimage of an interval under a measura

Updated Apr 18

Nowhere Dense Set

Analysis

How "thin" can a subset of a topological space be? The question has two distinct answers depending on whether one measures size by measure or by topology. A set of Lebesgue measure zero can still be dense — the [rationals]

Updated Apr 18

Total Boundedness

Analysis

In a metric space, boundedness is one of the simplest conditions one can impose: a set is bounded if it fits inside a ball of finite radius. In finite-dimensional Euclidean space $\mathbb{R}^n$, boundedness (together with closedness) suffices to characterise [compactness](/pa

Updated Apr 18

Equicontinuity

Analysis

In analysis, one frequently encounters sequences of functions and needs to extract convergent subsequences. For sequences of real numbers, the Bolzano--Weierstrass theorem guarantees that every bounded sequence has a convergent subsequence. For sequences of continuous functions, however, boundedness

Updated Apr 18

Baire Category Theorem

Analysis

How large can a "small" set be? More precisely: if we decompose a space into countably many pieces, each of which is "thin" in the sense that it is dense nowhere, can these thin pieces exhaust the entire space? In finite-dimensional Euclidean spaces and in the function spaces of analysis, intuition

Updated Apr 18

Tychonoff's Theorem

Analysis

In finite-dimensional analysis, compactness is the property that makes existence theorems work: the Extreme Value Theorem, the Bolzano-Weierstrass theorem, and the direct method in the calculus of variations all depend on extracting convergent subsequences from bounded collections. The natural quest

Updated Apr 18

Second Countable Space

Analysis

In a general topological space, the topology can be extraordinarily large --- the collection of open sets may have cardinality far exceeding that of the underlying set. This abundance creates serious obstacles. When we attempt to extract convergent subsequences, construct countable

Updated Apr 18

Locally Compact Space

Analysis

Many of the most important spaces in analysis and geometry are not compact — $\mathbb{R}^n$, smooth manifolds, locally compact groups — yet they still support a surprisingly rich theory of integration, function spaces, and duality. The Extreme Value Theorem fails on $\mathbb

Updated Apr 18

Cambridge IA Probability

Probability & Statistics

Probability is the mathematical framework for reasoning about uncertainty. At its heart lies a deceptively simple question: given an experiment whose outcome we cannot predict with certainty, how do we assign a consistent numerical measure to the likelihood of different events? The challenge is not merely philosophical — it is structural. We need…

Updated Apr 18

Path Connectedness

Analysis

A connected space cannot be split into two disjoint open pieces. But this algebraic condition — the absence of a separation — does not guarantee that you can "walk" continuously from one point to another. The topologist's sine curve is connected (it cannot be partitioned into

Updated Apr 18

Sequential Compactness

Analysis

The Bolzano-Weierstrass theorem is one of the most powerful results in elementary analysis: every bounded sequence in $\mathbb{R}^n$ has a convergent subsequence. This single statement underpins the Extreme Value Theorem, the existence of minimisers in optimisation

Updated Apr 18

Banach-Alaoglu Theorem

Analysis

In finite-dimensional analysis, compactness is available essentially for free: the Heine-Borel theorem guarantees that every closed and bounded subset of $\mathbb{R}^n$ is compact, and the Bolzano-Weierstrass theorem ensures that every bounded sequence has a convergent subsequence. These facts under

Updated Apr 18

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Measure Space

Nowhere Dense Set

Total Boundedness

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Second Countable Space

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Cambridge IA Probability

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Banach-Alaoglu Theorem

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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

Measure Space

Nowhere Dense Set

Total Boundedness

Equicontinuity

Baire Category Theorem

Tychonoff's Theorem

Second Countable Space

Locally Compact Space

Cambridge IA Probability

Path Connectedness

Sequential Compactness

Banach-Alaoglu Theorem