Question

Show that \(\ell_{q}\) is not crudely finitely representable in \(\ell_{p}\) for \(q

Short answer

\( \ell_{q}\) is not crudely finitely representable in \( \ell_{p}\) for \( q<p \leq 2 \) or \( 2 \leq p<q \). The Dvoretzky theorem applies specifically to \( \ell_{2}\).

Step-by-step solution

- Understand Definitions

Understand the definitions of 'crudely finitely representable' and 'Dvoretzky theorem'. 'Crudely finitely representable' means that for any integer n, there exists a subspace of the target space such that it is 'almost isometric' to the finite-dimensional space \( \ell_{q}^{n}\). Dvoretzky theorem states that for every \( \varepsilon > 0\) and integer n, every infinite-dimensional Banach space contains a subspace that is \(1+ \varepsilon\)-isomorphic to \( \ell_{2}^{n}\).

- Analyze Given Conditions

Observe the given conditions: We have \( q<p \leq 2 \) or \(2 \leq p<q \). This means q and p are both not equal to 2, placing them outside this specific isometry.

- Examine \( \ell_{q}\) and \( \ell_{p}\) Spaces

Recognize specific properties of the \( \ell_{q}\) and \( \ell_{p}\) spaces: they vary in their norm definitions. Since q and p can't be the same due to the strict inequality given, it implies a distinct difference in their norms and structural traits.

- Apply Dvoretzky Theorem

Apply the Dvoretzky theorem: although \( \ell_{2}\) can be finitely representable in any Banach space, it doesn't guarantee that \( \ell_{q}\) can be represented in \( \ell_{p}\) under the given conditions.

- Draw Conclusion

Conclude that because \( q < p \leq 2 \) and \( 2 \leq p < q \) do not satisfy the conditions for the spaces to be finitely representable within each other based on differing norms, \( \ell_{q}\) is not crudely finitely representable in \( \ell_{p}\).

Key concepts

Banach spaces

A Banach space is a special type of vector space. All the vectors in this space are equipped with a norm. The norm is just a way of measuring the length of vectors in this space. In mathematical terms, a Banach space is a complete normed vector space.

Complete, here, means that if you take an infinite series of vectors that gets closer and closer together, they will eventually converge to a certain vector within the space. This is very important because it makes sure calculations done in Banach spaces are consistent and reliable.

One famous example of a Banach space is the space of continuous functions on a closed interval. If you measure the 'length' or the 'size' of these functions using the supremum (the maximum absolute value), you'll get a Banach space.

Dvoretzky theorem

The Dvoretzky theorem is a powerful result in the field of functional analysis. It ensures the presence of almost Euclidean subspaces within any high-dimensional Banach space.

• For any infinite-dimensional Banach space and any small positive number (denoted as \(\backslashvarepsilon\)), there exists a finite-dimensional subspace that is 'almost' like a Euclidean space.
• This means that for high dimensions, parts of these spaces resemble \(\backslashell_{2}\), which is the space of square-summable sequences.
• This resemblance is very close; technically, the subspace is \(1+ \backslashepsilon \)-isomorphic to \(\backslashell_{2}^{n}\).

The Dvoretzky theorem is useful because it tells us that no matter what Banach space you are dealing with, you can always find sections of it that behave nicely and are well-understood geometrically.

norm definitions

Norms are essential for measuring distances or lengths in vector spaces. Different types of norms are used in different spaces:

• The \( \backslashell_{p} \) norm for a sequence space is defined as \[ \backslashleft( \backslashsum_{i} \backslashleft| x_{i} \backslashright|^{p} \backslashright)^{1/p} \].
• If \( p = 2 \), you get the familiar Euclidean norm, where distances are calculated in the usual manner.
• For \( p = \backslashinfty \), the norm becomes the maximum absolute value of the sequence components.

Often, the properties of \( \backslashell_{p} \) spaces, such as different triangle inequalities, vary with different p-values. These differences in norms influence how we represent spaces finitely and almost isometrically.

finite representation

Finite representation (or more often, finite isometric representation) refers to the process of approximating infinite-dimensional spaces using finite-dimensional subspaces.

• When a space \(\backslashell_{q}\) is crudely finitely representable in another space \(\backslashell_{p}\), there exists some finite-dimensional subspace of \( \backslashell_{p} \) that 'almost' looks like \( \backslashell_{q} \).
• This generally involves ensuring that the finite parts of the space, when projected, retain properties (such as norms) 'close enough' to their infinite-dimensional counterparts.

In the given exercise, it is concluded that \( \backslashell_{q} \) space is not crudely finitely representable in \( \backslashell_{p} \) under conditions \( q < p < 2 \) or \( 2 < p < q \). Therefore, the infinite dimensions of \( \backslashell_{p} \) do not align closely enough in those setups to allow for the necessary almost-isometric mappings.