Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Complete the inclusion:

??? c //\\t:-T A[t] u B[t]

??? c //\\t:-T A[t] u B[t]

( //\\t:-T A[t] ) u ( //\\t:-T B[t] ) c //\\t:-T A[t] u
B[t]

Decide if it's true:

//\\t:-T A[t] u B[t] c ( //\\t:-T A[t] ) u ( //\\t:-T B[t] )

//\\t:-T A[t] u B[t] c ( //\\t:-T A[t] ) u ( //\\t:-T B[t] )

It is false.

Decide if it's true:

( //\\t:-T A[t] ) u ( //\\t:-T B[t] ) c //\\t:-T A[t] u B[t]

( //\\t:-T A[t] ) u ( //\\t:-T B[t] ) c //\\t:-T A[t] u B[t]

It is true.

Let V be a vector space over |R or over |C.

f : V -> |R

What does it mean that f is a norm?

f : V -> |R

What does it mean that f is a norm?

(1) /\x:-V [ x=O <=> f(x) = 0 ]

(2) /\x,y:-V [ f(x+y) <= f(x) + f(y) ]

(3) /\a:-K /\x:-V [ f(a*x) = |a|*f(x) ]

K is |R or |C

(2) /\x,y:-V [ f(x+y) <= f(x) + f(y) ]

(3) /\a:-K /\x:-V [ f(a*x) = |a|*f(x) ]

K is |R or |C

Let V be a normed vector space.

Prove that /\x:-V ||x|| >= 0.

Prove that /\x:-V ||x|| >= 0.

||x|| = 1/2 * [ ||0-x|| + ||x-0|| ] >= 1/2 * ||0-0|| = 0

Let V be a normed vector space.

Prove that /\x,y,z:-V ||x-y|| <= ||x-z|| + ||z-y||.

Prove that /\x,y,z:-V ||x-y|| <= ||x-z|| + ||z-y||.

hint: x-y = (x-z)+(z-y)

Let V be a normed vector space.

Prove that /\x:-V ||-x|| = ||x||.

Prove that /\x:-V ||-x|| = ||x||.

Recall that -x = (-1)*x

Let V be a normed vector space.

Prove that /\x,y:-V ||x-y|| <= ||x|| + ||y||.

Prove that /\x,y:-V ||x-y|| <= ||x|| + ||y||.

||x-y|| = ||x+(-y)|| <= ||x|| + ||-y|| =

= ||x|| + ||(-1)*y|| = ||x|| + |-1|*||y|| = ||x|| + ||y||

= ||x|| + ||(-1)*y|| = ||x|| + |-1|*||y|| = ||x|| + ||y||

Define the Euclidean norm on |R^n.

x = (x[1],x[2],...,x[n])

||x|| = sqrt( +(k=1 to k=n) x[k]*x[k] )

(In the proof that it is a norm we have to use the Schwarz inequality.)

page 39 in "Topologia", the first/old notebook

||x|| = sqrt( +(k=1 to k=n) x[k]*x[k] )

(In the proof that it is a norm we have to use the Schwarz inequality.)

page 39 in "Topologia", the first/old notebook

Define the max norm on |R^n and prove that it is a norm.

x = (x[1],x[2],...,x[n])

||x|| = max { |x[k]| : k:-{1,2,...,n} }.

||x|| = max { |x[k]| : k:-{1,2,...,n} }.