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

Let $K$ be a field, let $V$ be a vector space over $K$ with zero vector $0_{V}$, and let $W$ be a linear subspace of $V$. Then the following hold.

1. (Vector space) The set $W$, equipped with the restrictions to $W$ of the addition and the scalar multiplication of $V$, is a vector space over $K$. Its zero vector is $0_{V}$, and for $w\in W$ the additive inverse of $w$ in $W$ is its additive inverse $-w$ in $V$.

2. (Finite sums agree) Let $n$ be a natural number and let $w\in W^{n}$ be an $n$-tuple in $W$. Then the finite sum of $w$ formed in the vector space $W$ of claim 1 is equal to the finite sum of $w$ formed in $V$, the latter being the finite sum of $w$ regarded as an $n$-tuple in $V$.

3. (Inner product) Suppose $K$ is the field of complex numbers and $V$ together with $\langle\cdot,\cdot\rangle$ is a complex inner product space. Then the map assigning to each pair $w,w'$ of elements of $W$ the complex number $\langle w,w'\rangle$ is an inner product on the vector space $W$ of claim 1. The norm induced on $W$ by this inner product assigns to each $w\in W$ the same real number $\lVert w\rVert$ as the norm induced on $V$; and the metric on $W$ obtained from it as in claim 3 of The Induced Norm is a Norm, and Induces a Metric is the restriction to $W$ of the corresponding metric on $V$.