Multiplication by zero: $0 \cdot a = 0$

Depends on

• Field

Used by

• ∑_k<n+1binomnk = 2ⁿ, and ∑_k<n+1(-1)ᵏιbinomnk = 0 for n ≥ 1 Corollary
• The union of the two coordinate axes of F² is closed under scalar multiplication and is not closed under addition, so neither closure condition implies the other Counterexample
• Three lines in F² that meet pairwise only in 0 and whose sum is F² with decompositions that are not unique, so pairwise trivial intersection does not give a direct sum Counterexample
• Integer powers aᵐ Definition
• (0,1) + (2,3) = (2,4), with supremum 4 = sup(0,1) + sup(2,3) Example
• F^ℕ is a vector space and the eventually zero families form a linear subspace of it that is the span of the standard unit families Example
• In F³ the three coordinate lines are linear subspaces whose internal direct sum is F³, and F⁰ is the zero space Example
• sup{q ∈ ℚ : q > 0, q² < 2} = √2 in ℝ, and no supremum in ℚ Example
• Two planes in F³ whose sum is F³ and whose intersection is a line, computed explicitly Example
• FALSE: ∑_k<n+1(-1)ᵏιbinomnk = 0 for every n ∈ ℕ False statement
• FALSE: a^m/n := (a^1/n)ᵐ extends to negative bases False statement
• FALSE: every real number has a real square root False statement
• FALSE: the supremum of a set belongs to the set False statement
• FALSE: The union of two linear subspaces is a linear subspace False statement
• A field has no zero divisors: ab = 0 ⟹ a = 0 or b = 0 Lemma
• Bernoulli's inequality (1+x)ⁿ ≥ 1 + nx Lemma
• Laws of finite sums and finite products Lemma
• Laws of finite sums and products in ℕ, and ι(∑_k<n aₖ) = ∑_k<n ι(aₖ) Lemma
• Laws of rational exponents Lemma
• Sign rules for products and monotonicity of multiplication Lemma
• Sign rules for products: (-a)b = -(ab) and (-a)(-b) = ab Lemma
• Squaring is monotone on the nonnegatives Lemma
• Supremum of a scalar multiple Lemma
• Supremum of a sumset: sup(S + T) = sup S + sup T Lemma
• The sign of a product Proposition
• Existence and uniqueness of n-th roots: a unique a^1/n ≥ 0 with (a^1/n)ⁿ = a Theorem
• ℍ is a division ring that is not commutative, hence not a field: q⁻¹ = bar q / N(q) for q ≠ 0, while ij = k and ji = -k Theorem
• The arithmetic mean, geometric mean inequality Theorem
• The binomial theorem in ℝ: (x+y)ⁿ = ∑_k<n+1 ιbinomnk xᵏ y^ n-k Theorem
• The Cauchy-Schwarz inequality for finite sums Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 1 result over 1 level. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

• W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 1 (standard reference, not scraped)
• M. Spivak, Calculus, 4th ed., Ch. 1 (standard reference, not scraped)
• Janssen and Lindsey, Rings with Inquiry: Fields (standard reference, not scraped)