1. The Holder’s inequality states that: if \(1\leq p,q<\infty,\frac{1}{q}+\frac{1}{q}=1\) and if \(x_k,y_k,k=1,2,cdots,n\) are complex numnbers then

\(\left(\sum_{k=1}^{n}\left|x_k+y_k\right|^{p}\right)^{\frac{1}{p}\leq\left(\sum_{k=1}^{n}\left|x_k\right|^{p}\right)^{\frac{1}{p}+\left(\sum_{k=1}^{n}\left|y_k\right|^{p}\right)^{\frac{1}{p} \)

\(\left(\sum_{n=1}^{n}\left|x_k+y_k\right|^{p}\right)^{r}\leq\left(\sum_{k=1}^{n}\left|x_p\right|^{p}\right)^{\frac{1}{2}+\left(\sum_{k=1}^{n}\left|y_p\right|^{p}\right)^{\frac{1}{2} \)

—>> \(\left(\sum_{k=1}^{n}\left|x_k+y_k\right|^{p}\right)^{\frac{1}{p}\leq\left(\sum_{k=1}^{n}\left|x_k\right|^{p}\right)^{\frac{1}{p}+\left(\sum_{k=1}^{n}\left|y_k\right|^{q}\right)^{\frac{1}{q} \)

\(\left(\sum_{k=1}^{n}\left|x_k+y_k\right|^{p}\right)^{\frac{1}{p}\leq\left(\sum_{k=1}^{n}\left|x_p\right|\right)^{\frac{1}{p}+\left(\sum_{k=1}^{n}\left|y_p\right|\right)^{\frac{1}{q} \)

2. A sequence \({x_n}_{n=1}^{\infty}\) of points in a metric space \((E,d)\) is a Cauchy sequence if

for every \(\epsilon>0\), there exists a positive integer \(N\) such that \(x_n\in B(x,\epsilon)\) for all \(n\in N\) where \(B(x,r)={y\in E:d(y,x)<\epsilon}\)

\(X_n_k+1>X_n_k\) and \(n_k\geq k, k=1,2,cdots\) and \(n_k:\mathbb{N}\rightarrow\mathbb{N}

and only if its component sequence converges

—>> for any \(\epsilon>0\), there exists an integer \(N_0>0\) such that for all \(m,n>N_0\) we get that \(d(x_n,x_m)<epsilon\)

3. Let \((E,d)\) be a metric space and \(K\) a subset of \(E\). Then \(K\) is said to be connected if

and only if every subset of \(K\) are only closed

if and only if \(K\) is not the only nonempty set that is open and closed

—>> if and only if it is connected as a subspace

\(K\) is only open

4. If \((E,d)\)is metric space and \(x_0\in E\), then the open ball centred at \(x_0\) of radius \(r>0\) is given by the set

—>> \(B(x_0;r)={y\in E:d((x_0,y)<r}\)

\(B(x_0;r)={y\in E:d((x_0,y)\leq r}\)

\(S(x_0;r)={y\in E:d((x_0,y)\geq r}\)

\(S(x_0;r)={y\in E:d((x_0,y)=r}\)

5. The Euclidean metric on \(\mathbb{R}^n\) is defined as

\(d(x,y)=\sum_{i=1}^{n}\left|x_i-y_i\right|\)

—>> \(d_2(x,y)=\left(\sum_{i=1}^{n}\left|x_i-y_i\right|^2\right)^{\frac{1}{2}\)

\(d_{\infty}(x,y)=max_{1leq ileq n}\left{\left|x_i-y_i\right|\right}\)

\(d_{\infty}(x,y)=min_{1leq ileq n}\left{\left|x_i-y_i\right|^{\frac{1}{2}}\right}\)

6. let \(f:\mathbb{R}\rightarrow\mathbb{R}\) be defined by \(f(x)=\left{\begin{array}{rcl} x^2+1,&\mbox{if}&x\leq 0\\\frac{1}{2}(x+2),&\mbox{if}&x\geq 0\end{array}\right\), then \(f\) is

not continuous at \(x=0\)

not continuous on \(\mathbb{R}\)

—>> continuous on \(\mathbb{R}\)

continuous at \(x=0\)

7. Let \((X,d_x)\) and \((Y,d_Y)\) be arbitrary metric spaces. A mapping \(f:(X,d_x)\rightarrow (Y,d_Y)\) is called a strict contraction if

—>> there exist a constant \(k\in[0,1)\) such that \(d_Y(f(x),f(y))\leq kd_X(x,y)\) for all \(x,y\in X\)

there exist a constant \(k\in[0,1)\) such that \(d_Y(x,y)\leq kd_Y(x,y)\) for all \(x,y\in X\)

there exist a variable \(k\in\mathbb{N}\) such that \(d_X(f(x),f(y))\geq kd(x,y)\) for all \(x,y\in X\)

there exist a constant \(k\in\mathbb{N})\) such that \(d_Y(x,y)geq kd_X(x,y)\) for all \(x,y\in X\)

8. The discrete metric is defined as \(d_0:E\times E\rightarrow \mathbb{R}\) such that

\(d_0(x,y)=\left{\begin{array}{rcl} 1,&\mbox{if}&x\neq y\\-1,&\mbox{if}&x=y\end{array}\right\)

—>> \(d_0(x,y)=\left{\begin{array}{rcl}1,&\mbox{if}&x\neq y\\0,&\mbox{if}&x=y\end{array}\right\)

\(d_0(x,y)=\left{\begin{array}{rcl}0,&\mbox{if}&x\neq y\\-1,&\mbox{if}&x=y\end{array}\right\)

\(d_0(x,y)=\left{\begin{array}{rcl}1,&\mbox{if}&x\geq y\\-1,&\mbox{if}&x\leq y\end{array}\right\)

9. A metric space \((E,d)\) satisfies the following except

\(d(x,y)\leq d(x,z)+d(z,y)\) for all \(x,y,z\in E\)

\(d(x,y)= 0\)

—>> \(d(x,y)\leq 0\) for all \(x,y\in E\)

\(d(x,y)=d(y,x)\) for all \(x,y\in E\)

10. Let \((X,d_x)\) and \((Y,d_Y)\) be metric space and let \(f:D(f)\subset X\rightarrow Y\) where \(D(f)\) is the domain of \(f\), then \(f\) is continuous if

—>> given \(\epsilon>0\), there exist \(\delta>0\) such that if \(x\in D(f)\) and \(d_X(x,x_0)<\delta\), then \(d_Y(f(x),f(x_0))<\epsilon.

given \(\epsilon>0\), there exist \(\delta>0\) such that whenever \(d_2(x,a)<\delta\), it follows that \(|f(x)-f(a)|<\epsilon\)

given \(\epsilon>0\), there exist \(\delta>0\) such that whenever \(d_max(x,x_0)>\delta\), it follows that \(d_max_Y(f(x),f(x_0))<epsilon\)

given \(\epsilon>0\), there exist \(\delta>0\) such that \(d_x(x,x_0)<\epsilon\) then \(d_X(f9x),f(x_0))<\delta\)

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\(d(x,y)\leq d(x,z)+d(z,y)\) for all \(x,y,z\in E\)

\(d(x,y)= 0\)

—>> \(d(x,y)\leq 0\) for all \(x,y\in E\)

\(d(x,y)=d(y,x)\) for all \(x,y\in E\)

\(d_0(x,y)=\left{\begin{array}{rcl} 1,&\mbox{if}&x\neq y\\-1,&\mbox{if}&x=y\end{array}\right\)

—>> \(d_0(x,y)=\left{\begin{array}{rcl}1,&\mbox{if}&x\neq y\\0,&\mbox{if}&x=y\end{array}\right\)

\(d_0(x,y)=\left{\begin{array}{rcl}0,&\mbox{if}&x\neq y\\-1,&\mbox{if}&x=y\end{array}\right\)

\(d_0(x,y)=\left{\begin{array}{rcl}1,&\mbox{if}&x\geq y\\-1,&\mbox{if}&x\leq y\end{array}\right\)

—>> there exist a constant \(k\in[0,1)\) such that \(d_Y(f(x),f(y))\leq kd_X(x,y)\) for all \(x,y\in X\)

there exist a constant \(k\in[0,1)\) such that \(d_Y(x,y)\leq kd_Y(x,y)\) for all \(x,y\in X\)

there exist a variable \(k\in\mathbb{N}\) such that \(d_X(f(x),f(y))\geq kd(x,y)\) for all \(x,y\in X\)

there exist a constant \(k\in\mathbb{N})\) such that \(d_Y(x,y)geq kd_X(x,y)\) for all \(x,y\in X\)

not continuous at \(x=0\)

not continuous on \(\mathbb{R}\)

—>> continuous on \(\mathbb{R}\)

continuous at \(x=0\)

\(d(x,y)=\sum_{i=1}^{n}\left|x_i-y_i\right|\)

—>> \(d_2(x,y)=\left(\sum_{i=1}^{n}\left|x_i-y_i\right|^2\right)^{\frac{1}{2}\)

\(d_{\infty}(x,y)=max_{1leq ileq n}\left{\left|x_i-y_i\right|\right}\)

\(d_{\infty}(x,y)=min_{1leq ileq n}\left{\left|x_i-y_i\right|^{\frac{1}{2}}\right}\)

—>> \(B(x_0;r)={y\in E:d((x_0,y)<r}\)

\(B(x_0;r)={y\in E:d((x_0,y)\leq r}\)

\(S(x_0;r)={y\in E:d((x_0,y)\geq r}\)

\(S(x_0;r)={y\in E:d((x_0,y)=r}\)

and only if every subset of \(K\) are only closed

if and only if \(K\) is not the only nonempty set that is open and closed

—>> if and only if it is connected as a subspace

\(K\) is only open

\(\left(\sum_{k=1}^{n}\left|x_k+y_k\right|^{p}\right)^{\frac{1}{p}\leq\left(\sum_{k=1}^{n}\left|x_k\right|^{p}\right)^{\frac{1}{p}+\left(\sum_{k=1}^{n}\left|y_k\right|^{p}\right)^{\frac{1}{p} \)

\(\left(\sum_{n=1}^{n}\left|x_k+y_k\right|^{p}\right)^{r}\leq\left(\sum_{k=1}^{n}\left|x_p\right|^{p}\right)^{\frac{1}{2}+\left(\sum_{k=1}^{n}\left|y_p\right|^{p}\right)^{\frac{1}{2} \)

—>> \(\left(\sum_{k=1}^{n}\left|x_k+y_k\right|^{p}\right)^{\frac{1}{p}\leq\left(\sum_{k=1}^{n}\left|x_k\right|^{p}\right)^{\frac{1}{p}+\left(\sum_{k=1}^{n}\left|y_k\right|^{q}\right)^{\frac{1}{q} \)

\(\left(\sum_{k=1}^{n}\left|x_k+y_k\right|^{p}\right)^{\frac{1}{p}\leq\left(\sum_{k=1}^{n}\left|x_p\right|\right)^{\frac{1}{p}+\left(\sum_{k=1}^{n}\left|y_p\right|\right)^{\frac{1}{q} \)

for every \(\epsilon>0\), there exists a positive integer \(N\) such that \(x_n\in B(x,\epsilon)\) for all \(n\in N\) where \(B(x,r)={y\in E:d(y,x)<\epsilon}\)

\(X_n_k+1>X_n_k\) and \(n_k\geq k, k=1,2,cdots\) and \(n_k:\mathbb{N}\rightarrow\mathbb{N}

and only if its component sequence converges

—>> for any \(\epsilon>0\), there exists an integer \(N_0>0\) such that for all \(m,n>N_0\) we get that \(d(x_n,x_m)<epsilon\)