Metrizable topological vector space

Key takeaways

Metrizable 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 Metrizable topological vector space to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Metrizable topological vector space from memory before moving on to harder problems.

Reference excerpt

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

Pseudometrics and metrics A pseudometric on a set X {\displaystyle X} is a map d : X × X → R {\displaystyle d:X\times X\rightarrow \mathbb {R} } satisfying the following properties:

d ( x , x ) = 0 for all x ∈ X {\displaystyle d(x,x)=0{\text{ for all }}x\in X} ; Symmetry: d ( x , y ) = d ( y , x ) for all x , y ∈ X {\displaystyle d(x,y)=d(y,x){\text{ for all }}x,y\in X} ; Subadditivity: d ( x , z ) ≤ d ( x , y ) + d ( y , z ) for all x , y , z ∈ X . {\displaystyle d(x,z)\leq d(x,y)+d(y,z){\text{ for all }}x,y,z\in X.}

A pseudometric is called a metric if it satisfies:

Identity of indiscernibles: for all x , y ∈ X , {\displaystyle x,y\in X,} if d ( x , y ) = 0 {\displaystyle d(x,y)=0} then x = y . {\displaystyle x=y.}

Ultrapseudometric A pseudometric d {\displaystyle d} on X {\displaystyle X} is called a ultrapseudometric or a strong pseudometric if it satisfies:

Strong/Ultrametric triangle inequality: d ( x , z ) ≤ max { d ( x , y ) , d ( y , z ) } for all x , y , z ∈ X . {\displaystyle d(x,z)\leq \max\{d(x,y),d(y,z)\}{\text{ for all }}x,y,z\in X.}

Pseudometric space A pseudometric space is a pair ( X , d ) {\displaystyle (X,d)} consisting of a set X {\displaystyle X} and a pseudometric d {\displaystyle d} on X {\displaystyle X} such that X {\displaystyle X} 's topology is identical to the topology on X {\displaystyle X} induced by d . {\displaystyle d.} We call a pseudometric space ( X , d ) {\displaystyle (X,d)} a metric space (resp. ultrapseudometric space) when d {\displaystyle d} is a metric (resp. ultrapseudometric).

Topology induced by a pseudometric If d {\displaystyle d} is a pseudometric on a set X {\displaystyle X} then collection of open balls:

B r ( z ) := { x ∈ X : d ( x , z ) < r } {\displaystyle B_{r}(z):=\{x\in X:d(x,z)<r\}} as z {\displaystyle z} ranges over X {\displaystyle X} and r > 0 {\displaystyle r>0} ranges over the positive real numbers, forms a basis for a topology on X {\displaystyle X} that is called the d {\displaystyle d} -topology or the pseudometric topology on X {\displaystyle X} induced by d . {\displaystyle d.}

Convention: If ( X , d ) {\displaystyle (X,d)} is a pseudometric space and X {\displaystyle X} is treated as a topological space, then unless indicated otherwise, it should be assumed that X {\displaystyle X} is endowed with the topology induced by d . {\displaystyle d.}

Pseudometrizable space A topological space ( X , τ ) {\displaystyle (X,\tau )} is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) d {\displaystyle d} on X {\displaystyle X} such that τ {\displaystyle \tau } is equal to the topology induced by d . {\displaystyle d.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Metrizable topological vector space

Start with the simplest possible case. Write down what Metrizable 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 Metrizable 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 Metrizable 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 Metrizable topological vector space

In research: Metrizable 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 Metrizable 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: Metrizable topological vector space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Metric spaces, Topological vector spaces, so understanding it makes those chapters shorter.
In everyday life: Look for Metrizable 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.

How to study Metrizable topological vector space in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Metrizable 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 Metrizable topological vector space out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Metrizable topological vector space in simple terms?

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

Why does Metrizable 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 Metrizable 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 Metrizable topological vector space.