Problem session / review for Sections 3.1–3.3

MATH 621 — Real Analysis I

Sections 3.1–3.3 — Norms, $\ell^p$ Spaces, and Induced Metrics (30 Problems)

How to use (pedagogy-first)

Progressive disclosure. Every problem has two nudges (Hint A → Hint B), an optional common pitfall, and a short solution outline. By default, all answers/outlines are hidden. Reveal only what you need, in order.
  • Self-explanation prompts ask you to verbalize the key step before you see a hint.
  • Interleaving: problems progress from definitions to induced metrics and geometry, revisiting earlier ideas.
  • Minimal guidance, not full solutions: outlines sketch the path, leaving details for you to fill in.
Tip: In class, use “Hint A” as a nudge. If multiple students stall, unlock “Hint B”; keep outlines for wrap‑up.

Quick recap (Sections 3.1–3.3)

Norm. $\|\cdot\|:X\to[0,\infty)$ with nonnegativity, definiteness ($\|x\|=0\Leftrightarrow x=0$), homogeneity ($\|cx\|=|c|\|x\|$), triangle inequality.

$\ell^p$. For $1\le p<\infty$: $\ell^p=\{x=(x_k): \|x\|_p=(\sum |x_k|^p)^{1/p}<\infty\}$; for $p=\infty$: $\ell^\infty=\{x: \sup_k|x_k|<\infty\}$.

Induced metric. $d(x,y)=\|x-y\|$ is a metric; norm balls are convex.

Problem set — 30 items (answers hidden; progressive hints)

Level 1 — Foundations (1–8)

Norm axiomsClassifyBloom: Understand

1) Spot the norms (on $\mathbb{R}^2$).

Decide which of the following are norms; justify succinctly: (a) $\|x\|=|x_1|+|x_2|$ (b) $\|x\|=\sqrt{|x_1|+|x_2|}$ (c) $\|x\|=\max\{|x_1|,|x_2|\}$ (d) $\|x\|=x_1^2+x_2^2$.

Hint A: Test homogeneity: does $\|cx\|=|c|\|x\|$ hold for all $c$?
Hint B: For (b), try $c=2$; for (d), try $c=-1$.
Pitfall: Definiteness means only the zero vector has norm $0$.
Outline: (a),(c) pass all axioms; (b) fails homogeneity; (d) fails homogeneity and triangle inequality.
ℓ¹Triangle inequalityBloom: Apply

2) Verify $\ell^1$ on $\mathbb{F}^d$.

Prove $\|x\|_1=\sum_{i=1}^d |x_i|$ is a norm.

Use $|u+v|\le |u|+|v|$ coordinatewise; sum over $i$.
Definiteness: if $\sum |x_i|=0$ then each $x_i=0$.
Check all four axioms; triangle is the only nontrivial one and follows from the pointwise inequality.
ℓ∞SupremumBloom: Apply

3) Sup-norm on bounded sequences.

On $\ell^\infty$, prove $\|x\|_\infty=\sup_k |x_k|$ is a norm.

Use $\sup|x_k+y_k|\le \sup|x_k|+\sup|y_k|$ for the triangle inequality.
Verify nonnegativity, definiteness, homogeneity; triangle via the sup inequality above.
ComputationFluency

4) Quick computations.

For $x=(-2,1,3)$ compute $\|x\|_1,\ \|x\|_2,\ \|x\|_\infty$.

Compute absolute values first; sum for $p=1$, max for $p=\infty$, and square–sum–root for $p=2$.
Enter your results before revealing anything:
Outline: $\|x\|_1=\sum|x_i|$; $\|x\|_\infty=\max|x_i|$; $\|x\|_2=\sqrt{\sum x_i^2}$.
GeometryUnit balls

5) Unit balls in $\mathbb{R}^2$.

Sketch $B_1(0)$ for $\|\cdot\|_1,\ \|\cdot\|_2,\ \|\cdot\|_\infty$ and label axes.

Translate inequalities to shapes: $\{|x_1|+|x_2|\le1\}$, $\{x_1^2+x_2^2\le1\}$, $\{\max(|x_1|,|x_2|)\le1\}$.
Expect diamond, circle, axis‑aligned square.
Outline: Argue by level sets of each norm and basic geometry.
Seminorm vs norm

6) A seminorm that isn’t a norm.

On $\mathbb{R}^2$ define $p(x)=|x_1|$. Show $p$ is a seminorm but not a norm.

Find a nonzero $x$ with $p(x)=0$.
Outline: Seminorm axioms hold; definiteness fails for $(0,1)$.
Weighted norms

7) Weighted $\ell^1$ (finite‑dim).

Let $\alpha_i>0$ and $\|x\|=\sum \alpha_i |x_i|$. Prove it’s a norm.

Triangle inequality is as in $\ell^1$; definiteness uses $\alpha_i>0$.
Outline: Check 4 axioms; only definiteness needs a sentence.
Weighted normsMax

8) Weighted sup‑norm.

Let $\beta_i>0$ and $\|x\|=\max_i \beta_i |x_i|$. Prove it’s a norm.

Use $\max(a_i+b_i)\le \max a_i + \max b_i$.
Outline: Homogeneity and definiteness are immediate from $\beta_i>0$; triangle via max inequality.

Level 2 — Core ℓ^p skills & induced metrics (9–16)

Series testMembership

9) Membership of $(1/n)$.

Decide $(1/n)_{n\ge1}\in\ell^p$ for $p=1,2,\infty$.

Use $p$‑series and boundedness.
Outline: Not in $\ell^1$; in $\ell^p$ for $p>1$; in $\ell^\infty$.
Series testMembership

10) Membership of $(1/\sqrt{n})$.

Decide $(1/\sqrt{n})\in\ell^p$ for $p=1,2,\infty$.

Compare $\sum n^{-p/2}$ to the $p$‑series threshold 1.
Outline: In $\ell^p$ iff $p>2$; also in $\ell^\infty$; not in $\ell^1,\ell^2$.
Induced metric

11) Induced metric.

Show $d(x,y)=\|x-y\|$ satisfies the metric axioms.

Symmetry: $\|x-y\|=\|y-x\|$ by homogeneity with $-1$.
Triangle: $\|x-z\|\le \|x-y\|+\|y-z\|$ by norm triangle.
Outline: Check nonnegativity/definiteness; symmetry; triangle via the norm inequality.
DistancesStandard basis

12) Distances in $\ell^p$.

In $\ell^p$, compute $\|\delta_m-\delta_n\|_p$ for $m\ne n$.

Only two coordinates are nonzero; evaluate the $p$‑norm.
Outline: $\|\delta_m-\delta_n\|_p=(|1|^p+|{-}1|^p)^{1/p}=2^{1/p}$; for $p=\infty$ the max is 1.
ConvergenceCoordinate control

13) Coordinatewise control.

In $\ell^p$ ($1\le p\le\infty$), show $\|x^{(k)}-x\|_p\to0\Rightarrow x^{(k)}_n\to x_n$ for each fixed $n$.

Bound a single coordinate by the whole norm.
Outline: $|x^{(k)}_n-x_n|\le\|x^{(k)}-x\|_p$ (or $\|\cdot\|_\infty$); take limits.
CounterexampleConvergence

14) Converse fails.

Give $x^{(k)}$ with $x^{(k)}_n\to0$ for each fixed $n$ but $\|x^{(k)}\|_p\nrightarrow 0$.

Try $x^{(k)}=\delta_k$.
Outline: Norm is constant $1$ for all $k$.
ContinuityInequalities

15) Norm difference is 1‑Lipschitz.

Prove $|\|x\|-\|y\||\le \|x-y\|$ for any norm.

Apply triangle to $x=(x-y)+y$ and swap roles.
Outline: Two applications of triangle yield both inequalities; combine.
Geometry

16) Translation & scaling of balls.

Show $B_r(x)=x+B_r(0)$ and $B_r(0)=r\,B_1(0)$ in any normed space.

Use $d(x,y)=\|x-y\|$, homogeneity, and translation invariance.
Outline: Unpack definitions directly.

Level 3 — Geometry & structure (17–24)

ConvexityBalls

17) Convexity of norm balls.

Show every open ball $B_r(x)$ is convex.

Fix $y,z\in B_r(x)$; consider $x-(ty+(1-t)z)$.
Apply triangle to $(1-t)(x-y)+t(x-z)$.
Outline: Conclude $\|x-(ty+(1-t)z)\|<(1-t)r+tr=r$.
Metric ≠ norm

18) A metric not induced by a norm (power metric on $\mathbb{R}^2$).

For $0<\alpha<1$, define $d_\alpha(x,y)=|x_1-y_1|^\alpha+|x_2-y_2|^\alpha$. Prove $d_\alpha$ is a metric, but not from any norm.

Use $(a+b)^\alpha\le a^\alpha+b^\alpha$ for triangle.
Unit ball fails convexity when $\alpha<1$.
Outline: Verify axioms; show non‑convex balls ⇒ not induced by a norm.
Powering a norm

19) $d_\alpha(x,y)=\|x-y\|^\alpha$ ($0<\alpha\le1$) is a metric; not a norm metric if $\alpha\ne1$.

Prove the claim.

Triangle from $(a+b)^\alpha\le a^\alpha+b^\alpha$; homogeneity would fail if it came from a norm.
Outline: Suppose $d_\alpha=\|\cdot\|’$; check $\|cx\|’$ vs $|c|\,\|x\|’$ leads to contradiction when $\alpha\ne1$.
Finite‑dim equivalences

20) $\|x\|_\infty \le \|x\|_p \le d^{1/p}\|x\|_\infty$ on $\mathbb{F}^d$.

Prove the inequalities for $1\le p<\infty$.

Left inequality: one coordinate attains the max; right: compare sum to $d\|x\|_\infty^p$.
Outline: Raise to $p$; apply bounds; take $p$‑th root.
Monotonicity in $p$

21) If $p\le q$ then $\|x\|_q\le \|x\|_p$ (finite $d$).

Prove the monotonicity.

Normalize by $\|x\|_\infty$ and use Problem 20.
Outline: Combine bounds to get the inequality.
AsymptoticsMembership

22) Sequence $x_k=k^{-\alpha}$: when is $x\in\ell^p$?

Determine all $(p,\alpha)$.

Use $p$‑series: $\sum k^{-\alpha p}$ converges iff $\alpha p>1$.
Outline: Translate to the classical series test.
Continuity

23) Norms are continuous.

Show $\|\cdot\|$ is continuous in its own induced metric.

Use Problem 15: the norm is 1‑Lipschitz.
Outline: Lipschitz ⇒ continuous.
Combining norms

24) Max and sum of norms are norms.

Given norms $\|\cdot\|_a,\|\cdot\|_b$, show $\max\{\|x\|_a,\|x\|_b\}$ and $\|x\|_a+\|x\|_b$ are norms.

Triangle: $\max(u+v)\le \max u + \max v$; and $(u_a+u_b)+(v_a+v_b)=(u_a+v_a)+(u_b+v_b)$.
Outline: Check 4 axioms for each construction.

Level 4 — Deeper/challenging (25–30)

ConvexityInequalities

25) Midpoint inequality.

Show $\big\|\frac{x+y}{2}\big\|\le \frac12(\|x\|+\|y\|)$ for any norm.

Write $x=\frac{(x+y)+(x-y)}{2}$ and apply triangle twice.
Outline: Rearrange to the desired inequality.
ℓ^p inequality

26) Clarkson‑type bound (basic).

For $1\le p\le\infty$, prove $\|x+y\|_p^p \le 2^{p-1}(\|x\|_p^p+\|y\|_p^p)$ (interpret $p=\infty$ as the triangle inequality).

Use $|a+b|^p \le 2^{p-1}(|a|^p+|b|^p)$ coordinatewise; then sum.
Outline: Apply inequality to each coordinate and aggregate.
Strict convexity

27) $\ell^p$ is strictly convex for $1

If $\|x\|_p=\|y\|_p=1$ and $x\ne y$, then $\|\tfrac{x+y}{2}\|_p<1$.

Use Minkowski and analyze equality cases.
Coordinatewise strict convexity of $t\mapsto |t|^p$ for $p>1$ is key.
Outline: Equality in Minkowski forces proportional vectors; with equal norms this would imply $x=y$.

Minkowski functional

28) Gauge of a balanced, convex, absorbing set.

Let $B\subset X$ be convex, balanced, absorbing, with $0\in\mathrm{int}(B)$. Show $p_B(x)=\inf\{t>0:x\in tB\}$ is a norm iff $B$ is symmetric and $p_B(x)=0\Rightarrow x=0$.

Homogeneity from balancedness: $cx\in |c|\,tB$.
Triangle from convexity: if $x\in sB$, $y\in tB$ then $x+y\in (s+t)B$.
Outline: Symmetry ensures $\|{-}x\|=\|x\|$; definiteness from interior and absorbing properties.
Classification

29) All norms on $\mathbb{R}$.

Show any norm on $\mathbb{R}$ has the form $\|x\|=c|x|$ with $c>0$.

Let $c=\|1\|$; use homogeneity and symmetry.
Outline: For $x\ge0$, $\|x\|=x\|1\|$; extend to $x<0$ via $\|{-}x\|=\|x\|$.
Quasi‑normsMetric ≠ norm

30) A metric on $\ell^p$ for $0

Define $d(x,y)=\sum_{k=1}^\infty |x_k-y_k|^p$ (finite sum domain). Show $d$ is a metric and explain why no norm induces it.

Triangle via $(a+b)^p\le a^p+b^p$ (since $p<1$).
Norm balls would be convex; here the balls fail convexity / homogeneity fails.
Outline: Verify metric axioms and use convexity argument to rule out norm‑inducibility.

Self‑assessment (exit ticket)

  • I can test whether a formula is a norm by checking all four axioms.
  • I can determine $\ell^p$ membership for standard sequences using $p$‑series.
  • I understand why $d(x,y)=\|x-y\|$ is a metric and why norm balls are convex.
  • I can give examples of metrics not induced by norms and explain why.
If any item is shaky, revisit just the Hints A/B first; save outlines for last.