mathematics

Locally convex topological vector space

Locally convex topological vector space is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Locally convex topological vector space rather than just read about it. In short: In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets.

Key takeaways

Locally convex topological vector space belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Locally convex topological vector space to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Locally convex topological vector space from memory before moving on to harder problems.

Reference excerpt

In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals. Fréchet spaces are locally convex topological vector spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm.

History Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1906 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced). After the notion of a general topological space was defined by Felix Hausdorff in 1914, although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces. Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him). A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like nets, the product topology and Tychonoff's theorem) to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spaces (in which case the unit ball of the dual is metrizable).

Definition Suppose X {\displaystyle X} is a vector space over K , {\displaystyle \mathbb {K} ,} a subfield of the complex numbers (normally C {\displaystyle \mathbb {C} } itself or R {\displaystyle \mathbb {R} } ). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

Definition via convex sets A topological vector space (TVS) is called locally convex if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets. The term locally convex topological vector space is sometimes shortened to locally convex space or LCTVS. In fact, every locally convex TVS has a neighborhood basis of the origin consisting of absolutely convex sets (that is, disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets. Every TVS has a neighborhood basis at the origin consisting of balanced sets, but only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both balanced and convex. It is possible for a TVS to have some neighborhoods of the origin that are convex and yet not be locally convex because it has no neighborhood basis at the origin consisting entirely of convex sets (that is, every neighborhood basis at the origin contains some non-convex set); for example, every non-locally convex TVS X {\displaystyle X} has itself (that is, X {\displaystyle X} ) as a convex neighborhood of the origin. Because translation is continuous (by definition of topological vector space), all translations are homeomorphisms, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector.

Definition via seminorms A seminorm on X {\displaystyle X} is a map p : X → R {\displaystyle p:X\to \mathbb {R} } such that

p {\displaystyle p} is nonnegative or positive semidefinite: p ( x ) ≥ 0 {\displaystyle p(x)\geq 0} ;

p {\displaystyle p} is positive homogeneous or positive scalable: p ( s x ) = | s | p ( x ) {\displaystyle p(sx)=|s|p(x)} for every scalar s . {\displaystyle s.} So, in particular, p ( 0 ) = 0 {\displaystyle p(0)=0} ;

p {\displaystyle p} is subadditive. It satisfies the triangle inequality: p ( x + y ) ≤ p ( x ) + p ( y ) . {\displaystyle p(x+y)\leq p(x)+p(y).}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Locally convex topological vector space

Start with the simplest possible case. Write down what Locally convex topological vector space claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Locally convex topological vector space before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Locally convex topological vector space ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Locally convex topological vector space

In research: Locally convex topological vector space appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Locally convex topological vector space in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Locally convex topological vector space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Convex analysis, Functional analysis, Topological vector spaces, so understanding it makes those chapters shorter.
In everyday life: Look for Locally convex topological vector space outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Locally convex topological vector space in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Locally convex topological vector space means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Locally convex topological vector space out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Locally convex topological vector space in simple terms?

Why does Locally convex topological vector space matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Locally convex topological vector space?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Locally convex topological vector space.