Polarization identity

Key takeaways

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

Reference excerpt

In linear algebra, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product. The norm associated with any inner product space satisfies the parallelogram law: ‖ x + y ‖ 2 + ‖ x − y ‖ 2 = 2 ‖ x ‖ 2 + 2 ‖ y ‖ 2 . {\displaystyle \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}.} In fact, as observed by John von Neumann, the parallelogram law characterizes those norms that arise from inner products. Given a normed space ( H , ‖ ⋅ ‖ ) {\displaystyle (H,\|\cdot \|)} , the parallelogram law holds for ‖ ⋅ ‖ {\displaystyle \|\cdot \|} if and only if there exists an inner product ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } on H {\displaystyle H} such that ‖ x ‖ 2 = ⟨ x , x ⟩ {\displaystyle \|x\|^{2}=\langle x,\ x\rangle } for all x ∈ H , {\displaystyle x\in H,} in which case this inner product is uniquely determined by the norm via the polarization identity.

Polarization identities Any inner product on a vector space induces a norm by the equation

‖ x ‖ = ⟨ x , x ⟩ . {\displaystyle \|x\|={\sqrt {\langle x,x\rangle }}.}

The polarization identities reverse this relationship, recovering the inner product from the norm. Every inner product satisfies:

‖ x + y ‖ 2 = ‖ x ‖ 2 + ‖ y ‖ 2 + 2 Re ⁡ ⟨ x , y ⟩ for all vectors x , y . {\displaystyle \|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}+2\operatorname {Re} \langle x,y\rangle \qquad {\text{ for all vectors }}x,y.}

Solving for Re ⁡ ⟨ x , y ⟩ {\displaystyle \operatorname {Re} \langle x,y\rangle } gives the formula Re ⁡ ⟨ x , y ⟩ = 1 2 ( ‖ x + y ‖ 2 − ‖ x ‖ 2 − ‖ y ‖ 2 ) . {\displaystyle \operatorname {Re} \langle x,y\rangle ={\frac {1}{2}}\left(\|x+y\|^{2}-\|x\|^{2}-\|y\|^{2}\right).} If the inner product is real then Re ⁡ ⟨ x , y ⟩ = ⟨ x , y ⟩ {\displaystyle \operatorname {Re} \langle x,y\rangle =\langle x,y\rangle } and this formula becomes a polarization identity for real inner products.

Real vector spaces If the vector space is over the real numbers then the polarization identities are:

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Polarization identity

Start with the simplest possible case. Write down what Polarization identity 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 Polarization identity 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 Polarization identity 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 Polarization identity

In research: Polarization identity 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 Polarization identity 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: Polarization identity is common in secondary-school and first-year university syllabi. It links to neighbouring topics Abstract algebra, Algebraic identities, Functional analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Polarization identity 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 Polarization identity in 20 minutes

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

Frequently asked questions

What is Polarization identity in simple terms?

In linear algebra, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in te…

Why does Polarization identity 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 Polarization identity?

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 Polarization identity.