12.1 Norms in Euclidean space[2CK]

(The fact that these are norms is demonstrated by the 332).

E330

[10D]Show that \(\lim _{p→∞}\| x\| _ p =\| x\| _∞\).

E330

[10F] Prerequisites:1.Having fixed \(t,s∈[1,∞]\) with \(s{\gt}t\) and \(x∈ℝ^ n\), show that \(\| x\| _{s}≤ \| x\| _{t}\). Also show that \(\| x\| _{s}{\lt} \| x\| _{t}\) if \(n≥ 2\) and \(x≠ 0\) and \(x\) is not a multiple of one of the vectors of the canonical basis \(e_ 1,\ldots e_ n\).

Hints:

1. use that \(1+t^ p≤ (1+t)^ p\) for \(p≥ 1\) and \(t≥ 0\); or

2. use Lagrange multipliers; or

3. remember that \(f(a+b){\gt} f(a)+f(b)\) when \(a≥ 0, b{\gt}0\) \(f(0)=0\) and \(f:[0,∞)→ℝ\) is strictly convex and continuous in 0 (see exercise 2), therefore derive \(\frac{d\hskip5.5pt}{d{t}}(\log \| x\| _{t})\) and set \(f(z)=z\log (z)\)).

Hidden solution: [UNACCESSIBLE UUID ’10G’] [UNACCESSIBLE UUID ’10H’]

[10J] Having fixed \(s,t∈[1,∞]\) with \(s{\lt}t\), show that \(n^{-1/s}\| x\| _ s≤ n^{-1/t} \| x\| _ t\) (where we agree that \(n^{-1/∞}=1\)). (Note that this is an inequality between averages).
(Hint. Set \(𝛼=t/s\) and \(y_ i=|x_ i|^ s\), then use the convexity of \(f(y)=y^{𝛼}\). Another tip: use 3.) Hidden solution: [UNACCESSIBLE UUID ’10K’] [10M] Let be given \(p,q∈[1,∞]\) such that \(1/p + 1/q = 1\)  ¹ and \(x,y∈ℝ^ n\); show the Hölder inequality in this form

In what cases is there equality?

Tips: Fix \(x_ i,y_ i≥ 0\). For the cases with \(p,q{\lt}∞\) you can:

• use Young inequality (3 or 3);

• use Lagrange multipliers;

• start from the case \(n=2\) and set \(x_ 2=t x_ 1\) and \(y_ 2=a y_ 1\); then, for cases \(n≥ 3\) use induction.

Hidden solution: [UNACCESSIBLE UUID ’10N’] [10P] Prerequisites:3.Infer the version

from ??. In which case does equality apply? [10Q] Prerequisites:3.Given \(p∈[1,∞]\) show the Minkowski inequality

There follows that \(\| x\| _ p\) are norms.

For \(p∈ (1,∞)\) find a simple condition (necessary and sufficient) that involves equality; compare it with 2; deduce that \(ℝ^ n\), with the norm \(\| ⋅\| _ p\) for \(p∈ (1,∞)\), is a strictly convex normed space (see 3). Hidden solution: [UNACCESSIBLE UUID ’10R’] [10S]Prerequisites:1,1,2.Let \(r{\gt}0\); if \(p∈[1,∞]\) then the ball \(B_ r^ p=\{ \| x\| _ p{\lt} r\} \) is convex; also \(B_ r^ p⊆ B_ r^{\tilde p}\) if \(\tilde p{\gt} p\). In the case \(n=2\) of planar balls, study graphically the shape of the balls as \(p\) varies. Are there points that are on the border of all balls? Hidden solution: [UNACCESSIBLE UUID ’10T’] [10V]If \(r{\gt}0\) and \(p∈(1,∞)\) then the sphere \(\{ \| x\| _ p= r\} \) is a regular surface. Hidden solution: [UNACCESSIBLE UUID ’10W’] [10X] Prerequisites:??.We equip \(ℝ^ n\) with the norm \(\| x \| _∞\): show that in dimension 2 the disk \(\{ x∈ℝ^ n , \| x\| _∞≤ 1\} \) is a square, and in dimension 3 it is a cube, etc etc.

Now we equip \(ℝ^ n\) with the norm \(\| x \| _ 1\): show that in dimension 2 the disk \(\{ x∈ℝ^ n , \| x\| _ 1≤ 1\} \) is a rhombus i.e. precisely a square rotated 45 degrees; and in dimension 3 the disk is an octahedron. [10Y]Find a norm in \(ℝ^ 2\) such that the ball is a regular polygon of \(n\) sides.

Hidden solution: [UNACCESSIBLE UUID ’10Z’]