Proof of A Linear Subspace is a Vector Space and Inherits an Inner Product

Claim 1. By Linear Subspace, $0_{V}\in W$, and $W$ is closed under the addition and the scalar multiplication of $V$; hence those operations restrict to maps from $W\times W$ to $W$ and from $K\times W$ to $W$, so the restricted operations are well defined on $W$.

Every axiom of Vector Space over a Field except the existence of a zero vector and of additive inverses is a universally quantified identity between elements formed from the two operations; each such identity holds for all elements of $W$ because it holds for all elements of $V$, and the operations of $W$ are the restrictions of those of $V$. For the zero vector: $0_{V}\in W$ and $0_{V}+w=w$ for every $w\in W$, so $0_{V}$ is a zero vector for $W$. For additive inverses: by Elementary Identities in a Vector Space the additive inverse $-w$ of $w$ in $V$ satisfies $-w=(-1)w$, so $-w\in W$ by closure under scalar multiplication, and $w+(-w)=0_{V}$. Hence $W$ with the restricted operations is a vector space over $K$, with zero vector $0_{V}$ and with the additive inverses of $V$.

Claim 2. Let $[n]$ be the initial segment determined by $n$; by the definition of $W^{n}$ the $n$-tuple $w$ is a map from $[n]$ to $W$ with values $w_{k}$, and since $W\subseteq V$ it is also a map from $[n]$ to $V$, which is what it means to regard $w$ as an $n$-tuple in $V$.

For $i$ a natural number with $i\in[n]$, write $\Sigma^{W}_{i}$ and $\Sigma^{V}_{i}$ for the finite sums $\sum_{k=1}^{i}w_{k}$ formed in the vector space $W$ of claim 1 and in $V$ respectively. Let $P(i)$ be the assertion: if $i\in[n]$, then $\Sigma^{W}_{i}=\Sigma^{V}_{i}$. We prove $P(i)$ for every natural number $i$ by induction.

For $i=1$, claim 1 of Properties of Finite Sums of Vectors, applied in $W$ and in $V$, gives $\Sigma^{W}_{1}=w_{1}=\Sigma^{V}_{1}$.

Assume $P(i)$ and suppose $S(i)\in[n]$, with $S$ the successor map of Natural Numbers. Then $i\in[n]$ by Properties of the Order on the Natural Numbers, as $1\le i$ and $i<S(i)\le n$. The recursion in claim 1 of Properties of Finite Sums of Vectors, applied in each of the two spaces, gives $\Sigma^{W}_{S(i)}=\Sigma^{W}_{i}+w_{S(i)}$ and $\Sigma^{V}_{S(i)}=\Sigma^{V}_{i}+w_{S(i)}$, where in the first identity the addition is that of $W$, which is the restriction of the addition of $V$ by claim 1. By $P(i)$ the two right-hand sides are equal, so $P(S(i))$ holds. Taking $i=n$ proves claim 2.

Claim 3. Conditions 1, 2 and 3 of Complex Inner Product Space are universally quantified conditions on elements of the space and on complex scalars, so they hold for the restricted map on $W$ because they hold on $V$, the operations of $W$ being restrictions of those of $V$ by claim 1. For condition 4, let $w\in W$: the number $\langle w,w\rangle$ is a nonnegative real number because this holds in $V$; and if $\langle w,w\rangle=0$ then $w=0_{V}$, which by claim 1 is the zero vector of $W$. Hence the restriction is an inner product on $W$.

Let $w\in W$. The norm induced on $W$ assigns to $w$ the nonnegative real number whose square is $\langle w,w\rangle$, and the norm induced on $V$ assigns to $w$ the nonnegative real number whose square is the same number $\langle w,w\rangle$; by Existence and Uniqueness of the Nonnegative Square Root these two numbers are equal, so the two norms agree at $w$.

Finally, let $w,w'\in W$. By claim 1 the element $w-w'=w+(-w')$ formed in $W$ is the same as the element formed in $V$, and by the previous paragraph the two induced norms agree on it. Hence the metric of claim 3 of The Induced Norm is a Norm, and Induces a Metric formed in $W$ assigns to the pair $(w,w')$ the same real number as the corresponding metric on $V$, that is, it is the restriction of the latter to $W$.