Dual Space of Linear Functionals.wmv

Linear functionals are functions that map vectors in an N dimensional vector space Vn to scalars. Linear functionals can be viewed as vectors. The linear space of functionals is called the dual space Vn*. Further, given a basis of Vn we can find a unique basis of Vn* called the dual basis. We can construct contravariant components as elements of Vn and covariant components as elements of Vn*. These constructions are useful in the theory of Manifolds and allow us to define tensors such as the metric tensor.

[supertren]

Thank you very much. Very clear explanation!.

[Mathview]

@KevinByrne2 Good questions. 1) We assume familiarity with linear spaces, linear dependence, independence, basis, dimension. These concepts are discussed earlier in this series. Basis vectors here have the usual meaning. One definition is: A basis of Vn is any set of linearly independent vectors that spans the linear space Vn.

2) The inner product is a separate construction. It follows from a metric. Many metrics are possible. A specific metric must be chosen to obtain an inner product space.

[Mathview]

What are covariant and contravariant vectors in an n-dimensional vector space?  In this video we discuss an algebraic theory of these things. First linear functionals are introduced mapping Vn to R. Then the collection of all such linear functionals is shown to be a linear space itself. This space is called the dual space to Vn denoted by Vn*.  There are many characterizations of the notion of contravariant and covariant vectors, of which this is one.

[Mathview]

@KevinByrne2 More on 2) Linear spaces need not have a metric (or an inner product.) A good example is the engineering concept of a state space. A state space is a non-geometric linear space. No orthogonality, no angles, no Euclidean metric or inner product, but its elements comprise a linear space, obey the laws of vector addition, scalar multiplication. The thermodynamic state space of a gas sample in a container with P, V, T, etc monitored is an example of such a linear space without metric.

[aSeaofTroubles]

Wow! So that's what this all means. THANK YOU!

[Mathview]

@KevinByrne2 Ah..yes TY. Allow me to amplify a bit... To be clear ... The "global" thermodynamic state space can be characterized as a differential manifold. Such manifolds support local linear spaces of differentials. So thermodynamic states are points in a manifold (not vectors) but also ARE vectors in local neighborhoods. That is, they are elements of local linear spaces of differentials. Such linear spaces exist in neighborhoods of every point of the manifold.

[Mathview]

@KevinByrne2 Linear spaces arise in thermodynamics as local linear spaces of differentials. The laws of thermodynamics are stated in terms of these differentials. All very interesting, and useful too.

[Mathview]

Two cool theorems on linear functionals (f) on topological vector spaces.
That is, f:V -> R where V is a topological vector space:
Th1: If f is continuous at a point x0 in V, then f is continuous on (all of) V.
Th2: Let f be a functional on a topological vector space V.
Then f is continuous on V if and only if f is bounded in some neighborhood of the zero vector of V.
Proof : Homework problem. hints: Both domain and range of f have useful topologies, and f obeys linearity. K&F pps.175, 176.

[shuewingtam]

At 22:10, you mistakenly wrote vector basis e sup k instead of e sub k. Dot product of covector basis e sup j and vector basis e sub j is kronecker delta.

[Mathview]

Linear spaces and the Dual Space of Linear Functionals. 

[pauluk]

Excellent presentation. Thank you. I got confused at slide 7 (8:00) where you said that a linear functional in V* evaluates to an an n-tuple (rather than the single scalar value I was expecting) but you later go on to point on slide 10 (11:40) the effect of such a functional on a PARTICULAR vector A in V is what gives that single value. And this seems to explain where the duality of the 'dual basis' comes in: we can 'pass' to the same element in the underlying field in either direction yes?

[nachoarroyo]

Hi dear @Mathview, I'm trying to understand the concept of RKHS in terms of what does the inner product mean in that sense, defined in a RKHS (given that such a concept does not mean the same thing as when we are referring to L_2). I found that the cencept of dual basis is too closely related.. Could you explain me the existent relationship please?
Thaks in advance...