Let S be any set. Carefully justify the validity of all the vector space axioms for the space ℱ(

Let S be any set. Carefully justify the validity of all the...

Key Concepts

Pointwise Operations

Pointwise operations refer to the method of defining the addition of two functions and the multiplication of a function by a scalar by evaluating them at each point in their domain. Specifically, for functions f and g from a set S to the real numbers, the sum (f+g)(s) is defined as f(s) + g(s) and, for a scalar c, (cf)(s) is defined as c * f(s). This approach is crucial in establishing that the set of all such functions forms a vector space because it ensures that the vector space axioms are satisfied at every point in the domain.

Function Spaces

Function spaces are collections of functions which themselves form vector spaces under suitably defined addition and scalar multiplication operations. In these spaces, the functions act as vectors and the operations are typically defined pointwise. Function spaces are significant in various areas of mathematics, including analysis, differential equations, and applied mathematics, as they allow the utilization of linear algebra techniques to study functional relationships and transformations.

Vector Space

A vector space is an algebraic structure formed by a collection of objects (called vectors) that can be added together and scaled by numbers (scalars). The structure must satisfy several axioms, such as associativity, commutativity of addition, existence of a zero vector, existence of additive inverses, and compatibility conditions with scalar multiplication. This concept is fundamental for solving linear equations, performing transformations, and studying systems in both pure and applied mathematics.

Vector Space Axioms

The vector space axioms provide a formal set of rules that any collection of objects must follow in order to be considered a vector space. These include closure under addition and scalar multiplication, associativity and commutativity of vector addition, distributive properties linking scalar and vector addition, the existence of an additive identity (zero vector), and the existence of additive inverses. Demonstrating that these axioms hold is essential in verifying that a given set equipped with defined operations qualifies as a vector space.

Transcript

00:01 So in this occasion we got this set of functions defined everywhere on the real line.

00:06 So basically are just functions from reels to reels.

00:10 That's what i have writing here.

00:12 But these functions satisfy a particular condition and is that if we avoid that functions on 1, that is equal to 0.

00:20 So we got a set of a function that satisfy this condition.

00:26 So we can write this space in this way.

00:29 And also we need to define two operations.

00:35 So some operation of functions are going to be defined in this way.

00:40 So with some two functions will be just fx plus gx.

00:46 And the scale multiplication is just taking the scalar kalar multiplying to the whole function.

00:53 So here is something important to verify and is that given this we need to verify the 10 axioms for a vector space to see if this space is a vector space.

01:10 So the first axiom is related to the closeness of the sum operation.

01:16 That means if we take two functions on this space, v, then the sum is also on v.

01:23 So how to check this? well, the first thing is really easy is if we have two functions that are defined on the real line and we sum them together, they are also going to be defined on the real line.

01:37 So that's one of the things.

01:39 But the other thing is that we need to check if the extra condition on the space is satisfied.

01:47 That means if the sum of these two functions evaluated at 1 is also equal to 0.

01:54 It's not that hard to prove that because we know how the sum of functions behave.

02:01 So f plus g at 1 is equal to f evaluated at 1 plus j at 1 and we know that these two functions are part of this space that means that they evaluate to 0 both so we got 0 plus 0 and this is again equals to 0.

02:21 So in general when we sum together two functions the sum is closed.

02:27 That means that the sum also lives in the space.

02:31 Then we need to prove the other axioms related with the sum so the next is the commutativity of this operation that means if we take f plus g in this case of x we know that this operation is defined as f of x plus g x and they are just functions real number they're mapping real numbers that means f take real numbers to real numbers so basically this evaluated at some point will be a number so these two values here will commute because it are real numbers so this will we can exchange them so we got g of x plus f of x and then we know that this based on how we define the sum operation on this space will be equal to j plus f of x so this axiom holds for this space the next is verify the associativity property that means now we got three functions so we need to check if they are equal if we agree them differently so f plus g plus h times x evaluate that x so we apply the definition of the sum operation on this part so we got we have f of x plus g plus h of x and then we apply again how is the definition of the sum operation in the space so we got f of x plus g of x and plus h of x the right hand side now is this.

04:42 We agree with f plus g plus h of x.

04:48 So we apply here the definition of the summation under the space.

04:51 So we got f plus g x plus hx.

04:58 And then here we apply again the definition of the sum operation, and we got f of x plus g of x plus h of x plus h of x and as you can see they are both equal so this axiom also is satisfied the next is to define the no vector that means a function that avoids to zero in all the real line that i have defined this in this way actually the the zero vector should satisfy this condition if we sum these two functions did return the same function f but that is satisfied by this constant function that evaluates everywhere is equal to zero.

05:50 And you can verify that this function is also part of this space b because it is zero everywhere.

05:59 So that means if we evaluate at 1, this is also going to be 0, so it's satisfied the condition of this vector, of this space.

06:10 So let's verify if this condition is satisfied.

06:14 So we got f plus the zero function.

06:17 X and this is equal to f of x plus zero x but this is zero everywhere that means that this is just equals to f of x plus zero which is just f of x so the zero vector exists and is part of this space so rate.

06:47 The next step is identify if the inverse of these functions also lives on this space b.

06:57 So that means that for every function living on this space, the inverse is also living in this space and satisfy the condition that if we sum these two functions together, we obtain the new is your vector.

07:14 So f plus minus f of x, okay? and we're going to define minus f of x, this function here as just minus f x.

07:31 So just putting a minus sign here.

07:33 So this becomes after applying the definition of the summation is f of x plus minus f of x.

07:45 But this, as i mentioned, is defined this way, just putting the minus sign in front.

07:53 So we got f of x minus f of x and this is equal to 0 for every x on the real line.

08:04 But what is the definition is a function that evaluates to 0 everywhere.

08:09 But that is just the definition of the function 0x that is equals to 0 for every x.

08:17 On the reels.

08:19 And this function is unique.

08:21 That means that this f minus f of x is just the zero function.

08:29 That corresponds to the zero vector in this space.

08:34 So great.

08:36 It also holds the five axiom.

08:40 Now we start with the axioms related with the square product...

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