User:Matt Kwan/MATH5605

Contents

render math: http://checkmyworking.com/misc/mathjax-bookmarklet/

PS1

1.1

suppose not, $\lambda_i$ first nonzero coefficient. But then $\sum\ne0$ in $(c_i,c_{i+1})-1/10$, contradiction

1.2

1. functions $t\mapsto e^{2\pi int}$ orthogonal therefore lin. independent, so (limits of) linear combinations that are equal have equal coefficients
2. rationals dense
3. $|C(I)|\le|\mathbb F^{I\cap \mathbb Q}|=|\mathbb N^\mathbb N|=|\mathbb R|$, but $|C(I)|\ge|I|=|\mathbb R|$ by 1.1

1.3

union of a chain of lin. independent sets is itself lin. independent and is an upper bound for the chain, so there is a maximal lin. independent subset.

1.4

1. definition
2. $\|x_n+y_n-x-y\|\le\|x_n-x\|+\|y_n-y\|\to 0$
3. $\|\lambda x_n-\lambda x\|=\lambda\|x_n-x\|\to 0$

1.5

• suppose (i). $\|y_N-y_M\|\le\sum_{n=N\land M+1}^{N\lor M}\|x_n\|\le \sum_{n=N\land M+1}^\infty\|x_n\|\to0$ as $n,m\to\infty$ so $y_n$ is cauchy for (ii). For fixed $N$, let $M\to\infty$ for (iii) (norm is continuous).
• suppose (ii). Take Cauchy $x_n$. Can find $N_k$ so that $\|x_n-x_m\|<2^{-k}$ for $n,m\ge N_k$. Then $\sum \|x_{N_k}\|<\infty$ so $x_{N_k}$ converges and so does $x_n$

1.6

zzz

1.7

also have $C_2^{-1}\|w\|\le\|T^{-1}w\|\le C_1^{-1}\|w\|$. suppose $W$ complete, fix cauchy $v_n\in V$. $Tv_n$ is cauchy also so converges to $T_v$, so $v_n$ converges to $v$.

1.8

for any $v \in V$, take $v_n \in V_0$, $v_n \to v$. $v_n$ Cauchy implies $T_0v_n$ is Cauchy and has a limit in $W$ since $W$ is complete. define $Tv = \lim\ T_0v_n$.

1.9

after 1.8, need to show that $T$ is bijective and isometric. For isometry choose $V_0\owns v_n\to v$, use continuity of $T$ and norm to show $\|Tv\|=\|v\|$; Injectivity follows. For surjectivity choose $W_0\owns w_n\to w$, so $T^{-1}w_n$ is cauchy and converges to some $v$. By continuity of $T$, $Tv=w$.

2.1

induction: $(a^n+b^n)^{1/n}= a(1+(b/a)^n)^{1/n}\to a$ if $a\ge b$.

2.2

$p=1$: diamond, $p=2$: circle, $p=\infty$: square, interpolate between. For $p=2$, orthogonal group: by polarization we have $\langle Te_i,Te_j\rangle=\langle e_i,e_j\rangle$ so $T^*T=I$. For $p=\infty$, identify corners as the only set of 2^n points that are mutually 2-separated, so corners have to map to each other and by repeated midpoints, so do the centers of each face. So a linear isometry is a composition of a permutation matrix and a scalar matrix with $\pm1$ on the diagonal. Apparently it's the same for other $p$ but he didnt prove it, i assume its untestable.

2.3

differentiate $s^p/p+t^q/q-st$ with respect to $t$ for the first inequality. Then $\sum \le\sum (|u_j|^p/p+|v_j|^q/q)\le1$. I think we also require $u_j\bar v_j$ to all be in the same direction for equality. Normalize both vectors for the next part. Finally choose $v_j=|u_j/\|u_j\||^{p/q-1}u_j/\|u_j\|$.

2.4

http://en.wikipedia.org/wiki/Minkowski_inequality

2.5

for any $\varepsilon>0$ there is finite $S_0$ so that $\sum_{S_0}(f+g)$ is $\varepsilon$-close to the LHS and $\sum_{S_0}f+\sum_{S_0}g$ is $\varepsilon$-close to the RHS. But these are equal so LHS and RHS are $2\varepsilon$-close.

2.6

here's the proof for $p=1$: Suppose $\sum\|f_n\|<\infty$, then for each $s$, $|\sum f_n(s)|\le\sum\|f_n\|<\infty$. By switching the order of summation with a finite $S_0$ and taking sups, have $\sum f_n\in \ell^1$ and $\|\sum^\infty f_n-\sum^Nf_n\|\le\sum_{n=N+1}^\infty\|f_n\|\to0$

apparently $p>1$ is untestable

2.7

asdf

2.8

too hard

2.9

given $u,v$, midpoint of $u$ and $v$ is the unique point in the intersection of two closed balls with radius $\|u/2+v/2\|$, and $TB(v,r)=B(Tv,r)$ so midpoints are preserved by $T$. Define $\tilde T=T-T0$ which also preserves midpoints so that $2\tilde Tv=2\tilde T(2v/2+0/2)=\tilde T(2v)$ and $\tilde T(u+v)=T(\frac{2u+2v}2)=\frac{\tilde T(2u)+\tilde T(2v)}2=\tilde Tu+\tilde Tv$. Then, by the density of the rationals $\tilde T$ is linear so $T$ was affine.

PS2

1.1

http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality#Proof

1.2

same criterion. for linearity, note $\langle x,y\rangle=\langle x,y\rangle_{\mathbb R}+i\langle x,y\rangle_{\mathbb R}$, where $\langle \cdot ,\cdot \rangle_{\mathbb R}$ is the real polarization thingy; from the real case we have $\langle x+\lambda z ,y \rangle_{\mathbb R}=\langle x,y \rangle_{\mathbb R}+\lambda\langle z ,y \rangle_{\mathbb R}$ for real $\lambda$, if not real deal with real and imaginary parts separately, noting that $\langle iz,y\rangle_{\mathbb R}=i\langle z,y\rangle_{\mathbb R}$ directly

1.3

$\langle P_S(v),e_i\rangle=\langle v,e_i\rangle$ so $P_S$ is idempotent. $\langle P_S(u),v\rangle=\sum \langle u,e_i\rangle\langle e_i,v\rangle=\langle u,P_S(v)\rangle$. $\langle v-P_S(v),e_i\rangle=0$ for all $i$ and if $w$ differs in say the $e_i$-coefficient from $P_S(v)$ then $\langle w-v,e_i\rangle\ne0$.

1.4

orthogonalize $v_n$ with gram-schmidt so that $P$ is upper-triangular and $P^*GP=I$. Then $(P^{-1})^*P^{-1}=G$ (inverse of U-T matrix is U-T; sweep through columns inductively proving strictly lower-triangular elements are 0)

1.5

not necessarily; consider the region above the graph of $f(x)=1/x$ in the first quadrant and its reflection about the $x$-axis

from ex. 3.22 in the notes: need compactness on one set, say $S_1$. Find sequences $x_n,y_n$ with $x_n\in S_1$, $y_n\in S_2$, $\|x_n-y_n\|\to\inf$. By compactness $x_n$ has a convergent subsequence to $x$. By Lemma 3.20, $x$ has a closest point $y\in S_2$. Then $\|x-y\|\le\|x-y_{n_k}\|\le\|x-x_{n_k}\|+\|x_{n_k}-y_{n_k}\|\to\inf$

1.6

untestable

1.7

untestable

2.1

any linear functional $M$ on $H$ is of the form $M(v) = \langle v,w \rangle$ for some $w$. Cauchy Schwarz gives $\|M\| \leq \|w\|$, taking $v = w/\|w\|$ gives $\|M\| \geq \|w\|$, so $\|M\| = \|w\|$. Write $w = \underbrace{w_1}_{\in H_0} + \underbrace{w_2}_{\in H_0^\perp}$, then $\|M\| = \sqrt{\|w_1\|^2+\|w_2\|^2}$. then $M|_{H_0} = L$ means $L(v) = \langle v,w_1 \rangle$, so $\|M\|=\|L\|$ implies $\|w_2\| = 0$, then $M(v) = \langle v,w_1 \rangle$ for all $v \in V$.

2.2

Define $b = (1,0,1,0,\ldots), c = (0,1,0,1,\ldots)$. $c_0 + \mathbb{F}b + \mathbb{F}c$ is a subspace of $\ell^\infty$. Can define a linear functional on this subspace by $ L((a_n)_{n\in\mathbb{N}}) = a_0 + \lim a_{2n} - \lim a_{2n+1}$. Easy to check $L$ is linear. $L$ is continuous since for any $d_n = d_0 + \lambda b + \mu c$ in the subspace, where $d_0\in c_0$, we have $\| L(d_n) \| \leq |d_0 | + |\lambda| + |\mu| \leq 3 \| (d_n) \|_\infty$. By Hahn Banach, can extend this to a linear functional $M$ on all of $\ell^\infty$ that does the same job as $L$ when both of the limits exist, and with $\| M \| \leq 3$.

equality is attained with $(1,-1,1,-1,\dots)$

2.3

probably untestable

2.4

"conversely" is untestable

Since $(L_\alpha)_{\alpha \in A} \in U^\perp$, $L_\alpha(u) = 0$ for all $\alpha \in A$, $u\in U$. Then for all $u$, $L(u) = \lim_{\alpha \in A} L_\alpha(v) = 0$. Thus $L\in U^\perp$.

2.5

asdf

3.1

$\delta_n$ has no convergent subsequence

3.2

Consider $F:\mathbb B(0,1)^{\mathbb N}\to\ell^p:f\mapsto(n\mapsto f(n)/(n+1))$. If $f_\alpha\to f$ in product topology, that says $f_\alpha(n)-f(n)\to 0$for each $n$. Since $|f-f_\alpha|^p$ is dominated by the integrable function $n\mapsto \left(\frac2{n+1}\right)^p$, by DCT $\|Ff_\alpha-Ff\|_p\to0$ and $F$ is continuous (if $p>1$). Image of compact set (tychonoff) is compact. Not compact for $\ell^1$: consider the open cover of all balls $B(f,1)$ where $f$ has finite support. Any finite subcover has a maximum supported $n\in \mathbb Z$, so by divergence of harmonic sequence theres a function in the set not covered.

3.3

fix $\epsilon > 0$. choose $M$ so that $(\sum_{m \geq M+1} |g(m)|^q)^{1/q} < \frac{\epsilon}{2}$, then $\sum_{m \geq M+1} \leq \|f_n\|_p\frac{\epsilon}{2} \leq \frac{\epsilon}{2}$ for all $n$. to bound the lower half, for each $m \in \{1,\dots,M\}$, can choose $N_m$ so that for all $n \geq N_m$, $f_n(m) \le \frac{\epsilon}{2M\|g\|_q}$, let $N = \text{max}\{N_m\}$, then for all $n \geq N$, $\sum_{m \in \mathbb{N}} \leq \epsilon$.

Doesn't hold for $p=1$, consider $f_n=\delta_n$, $g=1$. Converse is true, set $g=\delta_m$ (obviously not $\|f_n\|=1$ though)

3.4

hilbert spaces are reflexive so weak and weak-star coincide, then banach-alaoglu. For the second part, note $\|v_{\alpha_\beta}-v\|^2=\|v_{\alpha_\beta}\|^2+\|v\|^2-2\Re\langle v_{\alpha_\beta},v\rangle$

4.1

$\{L^{-1}S\}$ is a subbasis of the weak topology

4.2

sum of continuous operators is continuous and scalar multiple of continuous operator is continuous, appeal to 4.1. last part untestable

4.3

untestable

4.4

if $V\cong V^{**}$ then $V^{***}\cong(V^{**})^*\cong V^*$. second part untestable

5.1

asdf

5.2

$T(U)$ is finite dimensional so therefore isomorphic to some $\mathbb F^n$; $\overline{TB(0,1)}\subseteq B(0,\|T\|)$ is closed and bounded; heine borel.

Next, $\| T_n - T \| \rightarrow 0 $ for compact $T_n$ means that for any $\epsilon/3$, we can find $N$ such that $\| T_N - T \| <\epsilon/3$. Since $T_N$ is compact, $\overline{T(B(0,1))}=T(B(0,1))$ is compact and thus can be covered by finitely many $\epsilon/3$-balls. Let $\{ Tu_1, \dots, Tu_M \}$ be the centres of these balls, and pick some arbitrary $v\in T(B(0,1))$ ie. $v = Tu$ for some $u\in B(0,1)$. Then

1. $\| v - T_N u \| = \| Tu - T_N u \| \leq \epsilon/3 $.
2. We can pick some $u_m$ such that $ \| T_N u - T_N u_m \| \leq \epsilon / 3$.
3. $ \| T_N u_m- Tu_m \| \leq \epsilon/3$.

Putting these together with the triangle inequality, have $\| v - Tu_m \| < \epsilon$. Thus, $\{ Tu_1, \dots, Tu_M \}$ is a finite covering of $T(B(0,1))$ using $\epsilon$-balls. This proves $T(B(0,1))$ is compact, hence $T$ also compact.

Completeness should be assumed.

5.3

$\overline{TB}$ compact, $T^*$ continuous; $T^*\overline{TB}=\overline{T^*\overline{TB}}\supseteq\overline{T^*TB}$ compact; $T^*T$ compact and self-adjoint; make orthonormal $e_n$ with $\lambda_n e_n=T^*Te_n$, $\lambda_n\to0$ so that $Te_n/\sqrt\lambda$ is an o.n. basis for $TU$.If $v$ not spanned by $Te_n$ then for all $u$, $\langle v,Tu\rangle=\langle v,T\sum\langle u,e_n\rangle e_n\rangle=0$ and $\langle T^*v,u\rangle=0$ and $T^*v=0$. If $v\in B$ then $v=\sum a_nTe_n/\lambda_n+v^\perp$, $\sum|a_n|^2\le1$, $v^\perp\perp e_n\forall n$. Define finite rank $T^*_n$ as $T^*n$ restricted to span($e_1,\dots,e_n$) so $\|(T^*-T^*_n)v\|\le \max_{i>n} \lambda_i$ and$\|T^*-T^*_n\|\to0$, $T^*$ compact.

5.4

untestable

6.1

$|\hat g(n)|\le \int_I|g(t)e^{2\pi int}|=\|g\|_1$. ($\le \|g\|_2\|1\|_2=\|g\|_2$ by Holder's)

For the second part, if $f_k\to f$ then $\hat f_k\to \hat f$ by the above. Choose $q_k\in c_{00}$ that $1/k$-approximates $\hat f_k$, then $q_k\to \hat f$ so $\hat f\in c_0$.

6.2

prove formula with geometric series formula then, for $k<t(2N+1)<k+1$ the ingegrand is at least $|\sin((2N+1)\pi t)|/(\pi(k+1))$ so $\int_I\ge \sum_{k=0}^{2N} c/(k+1)$ for $c=\int\sin$. Compare with $\int^N 1/x=\log N$.

6.3

suppose not; $\ell^1\to C(T):\hat f\mapsto f$ is bijective and continuous (and well-defined by M-test) so is open and there is $C$ so that $\|\hat f\|_1\le C\|f\|$. So, suffices to construct a sequence of bounded functions $\|f_k\|$ with $\|\hat f_k\|$ unbounded, try $D_N$ from 6.2.

6.4

asdf

7.1

asdf

Theorems

• Hahn Banach
• Open Mapping
• Closed Graph
• Uniform Boundedness
• Banach-Alaoglu

Proofs

1.13

notes

4.4

use the sum completeness criterion from PS1 1.5. Take some sequence $L_n\in V^*$ which has $\sum^\infty \|L_n\|<\infty$. For each $v$ we have $\sum^\infty |L_nv|\le\|v\|\sum^\infty \|L_n\|<\infty$ so by the completeness of $\mathbb F$, $\sum^\infty L_nv<\infty$ and we can define $\sum L_n$ pointwise. Linearity is immediate, continuity follows from Theorem 1.13. We have $|\sum_{n=N+1}^\infty L_n v|\le \sum_{n=N+1}^\infty \|L_n\|\to0$ uniformly for $v\in B(0,1)$, so $\|\sum^N L_n-\sum^\infty L_n\|\to 0$.

4.16

to show it's a norm: inf of non-negative things is non-negative. $v + U = 0 + U \iff v \in U \implies \|v+U\|_q = 0$. if $\|v+U\|_q = 0$, then for each $n \in N$ there is $u_n \in U$ such that $\|v+u_n\|<\frac{1}{n}$. then $-u_n \to v$, so $v \in U$ since $U$ is closed, i.e. $v + U = 0 + U$. for $\lambda \neq 0 \in \mathbb{F}$, we have $\inf_{u \in U} \|\lambda v + u\| = \inf_{u \in U} \|\lambda v + \lambda u\| = |\lambda|\inf_{u \in U} \|v + u\|$. triangle inequality done in notes.

completeness: http://planetmath.org/quotientsofbanachspacesbyclosedsubspacesarebanachspacesunderthequotientnorm

5.7

asdf

6.2

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6.4

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6.25

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6.27

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