[BOŞ_SES] Hello.

Now the number of these spaces, an additional transaction at a time

by identifying more useful, we are going to a richer space.

The process can be made more're going to space.

In a previous type of space, Euclidean space on linear spaces,

We also obtained the Euclidean space by defining domestic product.

But here we are, with our habit with the plane's vector length

We may have given him the impression that it is multiplied by.

In fact, the length of the vector, can be defined independently from the inner product.

But we have it, it also leads to a variety of spaces,

business do not want to mix it up, but,

length, defined by the length of the vector inner product,

ie generalization of the Pythagorean theorem, we know that in the vector in order.

If we define the long way, but it will be a Euclidean space in the interior

Multiplying tanımlayabilel I will be the linear space of Euclidean space I tanımlayabilel,

that occurs when such a hierarchy of spaces.

If the length of the inner product of a vector of length

the square root of its own so we define this space called Hilbert space.

There is also a Hilbert space concept completeness,

but you do not need to go into it too much.

It is fully significant.

So while the works in this space, also important not shifted out of the space.

We will not dwell on that, because we get in practice

just as well-defined spaces, will be fully defined.

We call them Hilbert space.

So this will be a Euclidean space, on a neck,

or function of the length of any vector inner product

We identify with him taking the square root, we call it the space will be complete.

Now, more than dweII on completeness of this concept.

The finite size

Euclidean space is full and they form spontaneously Hilbert spaces.

The challenge may be in space in order to understand the function of completeness.

To ensure that this completeness we deal with in our space,

they are going Hilbert space.

Generally, Hilbert spaces of finite size

L large square frame is shown here because we found.

Function spaces indicated by the small L frame.

These are the most commonly used type of space.

Identity of two vectors in the plane, we see rather inequality.

We know that the inner product of x and y x height,

multiplied by y length, it is multiplied by the cosine of theta.

But for the cosine of theta is smaller than that on the other hand, inner product,

It must be the product of small size.

In the plane, that we know from geometry.

But when we went to higher-dimensional space,

it also needs to be proven.

Yet we know the geometry of the plane; a one point

Go to another point along a straight line, so the vector x plus y

We know that the vector x to go plus the sum of the vector y go.

If we look at their height, the straight line connecting two points

We know that it is shorter than the broken line simple geometry.

All this, in Euclidean space,

Watching it can also be provided in Hilbert space.

Because there are tribes need to define the Hilbert space.

Inner product defined in Euclidean space.

The first of these, ie less than 1 cosine theta

Our simple geometric information that tells

As it is seen here as a generalization,

Schwarz inequality occurs.

There's proof.

The proof of this I will follow you to watch.

Not prove too difficult.

Applies to function spaces.

If we do that in terms of components, inner product,

Rn in the components,

xy domestic product was the sum of the product components.

It will be smaller than the product of the size of the individual vectors.

Length of the component as a generalization of the Pythagorean theorem

The sum of the square; for giving this long of the length,

for giving the square of the length, we take the square root.

Like for years.

R in space, and it turns Schwarz inequality.

This is proving very useful and there is also a cultural dimension,

because the cosine of theta is smaller than the plane,

How can broadly apply show.

Yet a broader application of the product of two functions that can be called

integral, the size of functions, smaller than the product of length.

This, showing the Schwarz inequality in function spaces.

These topics such as the convergence of the series is a method of plant-available form.

I told triangle inequality, we know from the plane.

Proof of the exceptionally easy.

Here are the two sides, Let the x plus y squared.

Height was given to internal cross to; What we want to find the size of the vector,

We take the inner product with himself.

When we open this size obtained individually.

Now, a little while ago, the Schwarz inequality point x y,

i.e. the product x y X.

We knew it was smaller than the size multiplied by y neck.

So, a larger domestic product for this thing to last place,

inequality will occur and the direction of the inequality will be.

In this square we find the right x height in square meters,

x plus y is the length of the frame's height.

Square roots lifted here, we find inequality in the theorem.

This in Euclidean space when expanded in a Hilbert space,

When we define the size with internal Carp

correct the length of this vector connecting two points,

It will be the length of the vector x plus y, single vector x plus

y vector is smaller than the sum of the lengths

that says that the longer the broken line.

This is not something new, but also be extended to its function

a feature that can be found exceptionally stunning.

Here is the triangle inequality in this way.

The length of a vector sum of squares of components.

We find the square root of the neck when you receive it.

Where x is the length of the vector, wherein the length of the vector y; i.e. a fracture

x along the length of the first line, then to y length

We know that the right to go directly to x plus y little neck.

In functions also have this feature.

This is still a lot of features such as convergence, related function

It is a proof of inequality that works in many features.

Now here we finish our this section.

I would advise you to reconsider your homework and do it.

This is now not extend beyond anything we have ever seen on the subject.

These features, as they solved these problems by,

You will better understand the issues that appear.

A summary is also given.

This summary is also, as I said before, a compilation of the issues we see.

It will revise the sections you know immediately identify.

The parts you know less, you can learn through the text back to the beginning.

But just not able to learn anything at all to see them.

But what you know and do not know what,

I think it's useful to know what you a reminder.

Such quite well for a long summary.

Came because quite a few new concepts,

because linear spaces, a new concept that we saw.

Now we finish here said.

The next topic is, until now the function spaces,

We only work with algebraic vectors,

We work essentially on RNA vectors.

Now a major issue in the function space, this Fourier series.

This function of how the plane in space i and j,

space i, j, k unit using the base vectors

We were able to show all kinds of vector; we see that cosine

and able to produce an orthogonal base of the sinus.

Also function using these very large

it is possible to define families.

This is like separating the components of a vector,

dedicating to the component functions.

The next main section, it will be.